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Washington (CNN) -- A small but growing group of House Republicans is increasingly worried about the fallout from the government shutdown and say it's time for Speaker John Boehner to allow a simple vote on a spending bill.
Defunding Obamacare can wait for now, they say.
"I'm trying to be optimistic but at the same time I have a really, really tough time when people are out of work and they can't pay their bills," Rep. Michael Grimm of New York told reporters Wednesday. "Though it might be a political loss for us ... this is an untenable situation."
Rep. Scott Rigell, whose Virginia district is home to a significant number of military members and civilian contractors, was one of the first to publicly break away.
GOP Rep.: You're beautiful but be honest
Sen. Reid: I will not pick and choose
The game is the same, but many of the players have changed. Congress and the president are facing off in another supreme spending showdown. If they don't agree on a funding bill by the end of September 30, much of government will shutdown. This last happened in 2011, when Congress avoided a shutdown by passing a spending measure shortly after the midnight deadline hit. Who controls what happens this time? Take a look at the key players who will determine how this fight ends:
-- From CNN Capitol Hill Reporter Lisa Desjardins. CNN's Deirdre Walsh and Ted Barrett contributed to this report.
Photos: Key players in the shutdown debate
The shutdown and national security
A U.S. Capitol police officer walks past a statue of Gerald Ford in the rotunda on Tuesday, October 1. The Capitol is closed to tours because of the government shutdown.
Parks, museums: Sorry, we're closed
"We fought the good fight," he said in a tweet on Tuesday, but acknowledged it was time to move on.
Boehner hosted small groups of concerned members on Wednesday. A spokesman for Boehner declined to talk about the sessions.
A Republican source familiar with one of Wednesday's meetings said Boehner listened, but didn't signal he was willing to allow a vote on a clean bill.
"They weren't strong-armed, and they weren't asked to step back," the source said of the moderates in the meeting. It was taken as a positive sign that Boehner wasn't trying to muzzle the effort.
Another House Republican source acknowledged that the group doesn't yet have the numbers, muscle or will to force Boehner's hand. To do so, they would need to stick together and vote with Democrats to block any piecemeal spending bills from coming up.
The only Republican to do that so far is Rep. Peter King of New York.
One of the Republicans who met with Boehner Wednesday told CNN they are giving him a bit more time to let things play out, but could decide to rebel by the end of the week.
The shutdown: Americans on the edge
White House meeting
There were no apparent breakthroughs during a midweek meeting at the White House between congressional leaders and President Barack Obama.
Descriptions of the meeting ran the gamut. Obama called the session "useful;" House Minority Leader Nancy Pelosi said it was "worthwhile" and Boehner cast it as a "polite conversation." Senate Minority Leader Mitch McConnell, however, called it "unproductive."
But the major players were all in the same room at the same time, talking to each other -- something that hasn't happened much in recent weeks.
Following Cruz's playbook
As the shutdown lingers, some Republican moderates are openly frustrated that tea party darling Sen. Ted Cruz of Texas appears to be calling the shots on what House Republicans do next. Cruz was one of the first to suggest passing narrow bills that fund those government agencies or functions that generate any public backlash.
"I think the leadership is committed to play the Cruz strategy all the way out," California Rep. Devin Nunes told reporters, before adding "if you can call it a strategy."
For two days, GOP leaders have pushed through a series of piecemeal spending bills for floor votes that would fund things like veterans affairs, national parks and medical research. House Majority Leader Eric Cantor said Wednesday they plan to continue doing this.
"We've got ways to ease the pain on people," Cantor said. "We agree on a lot around here. We ought to fund that, and then we ought to sit down and talk about that which we don't."
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Boehner: Democrats will not negotiate
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A Park Service police officer stands guard in front of the Lincoln Memorial during a partial shutdown of the federal government in November 1995. Many government services and agencies were closed at the end of 1995 and beginning of 1996 as President Bill Clinton battled a Republican-led Congress over spending levels.
Photos: The last government shutdown
Still, the spending measures have no hope of passing, because the Democratic-led Senate won't approve the bills and, even if they did, the White House has promised a veto.
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Moderates meet
King hosted a group of mostly moderate GOP members in his office early Wednesday that want Boehner to allow a vote on a clean spending bill. He told reporters about 10 members attended, but said he believes there are about two dozen who would publicly back a clean spending plan -- one that doesn't try to strip the funding from President Barack Obama's signature healthcare program.
"I could be wrong, but I think if you had a secret ballot, 180 would vote for a clean CR," King said.
But it's more likely that a shift in House GOP strategy won't come in public defiance on the House floor, King said, but in quiet talks behind closed doors.
"Maybe it's because I come from New York. I rely on back room meetings to get things done," he said. "I'm hoping someone's going to meet behind the scenes somewhere and we're going to make a deal."
One senior Republican familiar with the talks argued that the effort may be small now, but it is expanding, and will grow as more Republicans hear from constituents back home that are hurting from the shutdown.
"It's Day 2 of the shutdown -- we went from six or seven (members) to over 20 today," the senior Republican told CNN.
Another GOP member familiar with the discussions told CNN they would only get serious if they stood together as a group to block a vote.
"The only way we're going to get Boehner and Cantor to change course is if we can bring things to a halt," said the source, who asked to speak anonymously while talks continue.
A perilous strategy
But it could be risky for these House Republicans to take a stand against the tea party faction of the GOP.
At the weekly lunch of the Republican Study Committee, a group of fiscal conservatives, the rumblings of the moderate GOP members came up. Some in the room said they should "go after" those fellow Republicans and put pressure on them to fall in line, according to a GOP source familiar with the discussions.
But another Republican congressional source in the meeting said the message was softer.
Members of the committee were encouraged to have "one-one-one converstations" with moderates to convince them to stick with the current GOP leadership strategy.
Nunes told reporters he expected the shutdown to go through the weekend and possibly through mid-October when Congress needs to raise the debt ceiling. He doesn't think the current House Republican plan, which he repeatedly said is being dictated by Cruz, is helping the GOP cause of defunding or delaying Obamacare.
But he said he will vote for the smaller spending bills out of loyalty to Boehner, even as he criticized the group behind Cruz as "lemmings."
"I'm going to continue to support our leadership. Even if we have entered the valley of death, when you enter the valley of death you have to keep running and the whole team has to stick together," a frustrated Nunes told reporters outside the House floor.
King acknowledged the effort to get more Republicans to push for a clean spending bill could take some time and probably wouldn't result in a new strategy until "the tea party has had enough."
U.S. shutdown threatens launch of NASA's next mission to Mars
CNN's Dana Bash and Alan Silverleib contributed to this story.
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Rep. Mark Meadows, R-North Carolina -- The architect. During Congress' August recess, the tea party-backed freshman wrote to Republican leaders suggesting that they tie dismantling Obamacare to the funding bill. Though initially rejected by GOP leadership, 79 of Meadows' House colleagues signed on to the letter, which quoted James Madison writing in the Federalist Papers, "the power over the purse may, in fact, be regarded as the most complete and effectual weapon ... for obtaining a redress of every grievance."
Speaker John Boehner, R-Ohio -- The coach. He'll make the key play call. The top Republican leader in the land may be the most important player in the days immediately before a possible shutdown. Boehner could decide whether to push through the Senate's version of a spending bill and keep government running, or he could float a third version with some other Republican wish list items in it. If he takes the second option, Boehner could risk a shutdown but could also force the Senate into a tough position: give House Republicans something or send federal workers home. Timing on all this will be critical.
Sen. Ted Cruz, R-Texas -- The revolutionary or rabble rouser, depending on your viewpoint. The tea party firebrand could lead a long filibuster on the Senate floor, delaying passage of a spending bill until just one day before the deadline on Monday, September 30. Cruz has stoked the anti-Obamacare flames all summer, but recently angered fellow Republicans by openly saying that the Senate does not have the votes to repeal the health care law.
Sen. Marco Rubio, R-Florida -- Senator to watch. The potential presidential candidate has been one of three senators (Cruz and Mike Lee, R-Utah, being the others) pushing to use the government shutdown debate as a way to repeal or defund Obamacare. But watch his actions and language as a shutdown nears to see if he digs in or if downshifts at all.
Sen. Harry Reid, D-Nevada -- The man steering the ship in the Senate. Master at using Senate procedure to his advantage, Reid is the main force in controlling the voting process in the chamber and ensuring that an attempted filibuster by tea party-types fails. The majority leader will be a primary negotiator if we reach phase three, if the House does not accept the Senate spending bill.
Sen. Mitch McConnell, R-Kentucky -- If Reid steers the ship, McConnell controls the headwinds. Which is good news for Reid, at least initially. The Republican leader and several of his members say they will vote against Cruz's filibuster and in favor of a spending bill with no limits on Obamacare. Meaning, in favor of a bill that just funds government. McConnell generally has been leery of running into a shutdown or default. In fact, one legislative method for avoiding default is named after him.
Sen. Patty Murray, D-Washington -- The consigliore. Murray, center, does not seek the outside limelight, but the Senate Budget Committee chairwoman is a major fiscal force behind the scenes on Capitol Hill. Known by fellow Democrats as a straight shooter, she is also an experienced negotiator, having co-chaired the laborious, somewhat torturous and unsuccessful Super Committee.
Rep. Tom Graves, R-Georgia -- The new militia leader. The freshman congressman from Georgia, second from right, is one reason the debate has reached this point. Graves led the charge that blocked the original proposal by House Republican leaders. That would have kept government funded and had a detachable portion on Obamacare. Instead Graves and other conservatives forced their leaders to pass a spending bill with a mandatory defunding of Obamacare.
Rep. Peter King, R-New York -- The blunt statesman. King is outspoken against many tea party tactics, calling the move to tie Obamacare to the must-pass spending bill essentially a suicide mission and Cruz "a fraud." He is pushing for Republicans to accept a more "clean" spending bill that can pass the Senate and avoid a shutdown.
Thomas Donohue, president and CEO of the U.S. Chamber of Commerce -- The heavy. Donohue is known for his deep connections and his aggressive lobbying on behalf of business. He and the Chamber are urging Republican lawmakers to avoid a shutdown. The Chamber is an important political backer for conservatives, but has had mixed success with the current Congress, locking in firm anti-tax positions but unable to push through immigration reform so far.
Michael Needham, president of Heritage Action -- The driving force. Needham runs the political offshoot of the conservative Heritage Foundation and has been unrelenting in urging lawmakers to repeal Obamacare. He has told Republicans not to fear a potential shutdown, saying they would suffer more politically from allowing Obamacare to continue.
President Barack Obama -- The campaigner and CEO. Expect the president to use his podium more as a shutdown nears, aiming at public opinion as Democrats in Congress position themselves. If House Republicans send back a new proposal close to the September 30 deadline, the president and Democrats will have to decide what move to make next.
Rep. Eric Cantor, R-Virginia -- The powerful lieutenant. Cantor, the House Republican No. 2, is much more closely allied with conservatives and tea party members in the House than is Speaker Boehner. The two have not always agreed on every strategy during potential shutdown debates, but have been in public lockstep during the current go-around.
Rep. Nancy Pelosi, D-California, and Steny Hoyer, D-Maryland -- Players on deck. The top two House Democrats are mostly watching and waiting. But they will play a critical role once Boehner decides his next move. They could either bring Democratic votes on board a deal or be the loudest voices against a new Republican alternative. Hoyer will be interesting to watch; he has strongly opposed both the House and Senate plans as cutting too much in spending.
Rep. Kevin McCarthy, R-California -- The numbers guy. McCarthy, the House whip, has the tricky job of assessing exactly where Republican members stand and getting the 217 votes it takes to pass a bill in the chamber. He is known for his outreach to and connection with many of the freshmen House members who align with the tea party.
Rep. Paul Ryan, R-Wisconsin -- Member to watch. The vote of the House budget chairman and former vice presidential nominee is an important signal both within Republican ranks and to the public at large. Ryan has voted against some funding measures in the past, including the emergency aid for Superstorm Sandy recovery. But he was a "yes" on the last extension of the debt ceiling.
Rep. Ileana Ros-Lehtinen, R-Florida -- Another member to watch. A former committee chairwoman (Republican rules have term limits for committee chairs), Ros-Lehtinen knows House politics and procedure inside out. Depending on the issue, she has been described as a conservative or moderate, and occasionally as a libertarian.
Barricades around the World War II Memorial in Washington prevent people from entering the monument on October 1. The memorial was temporary opened to veteran groups who arrived on Honor Flights on a day trip to visit the nation's capital.
World War II veteran Russell Tucker of Meridian, Mississippi, stands outside the barricade as he visits the World War II Memorial in Washington on October 1.
World War II Veteran George Bloss, from Gulfport, Mississippi, looks out over the National World War II Memorial in Washington, on October 1. Veterans who had traveled from across the country were allowed to visit the National World War II Memorial after it had been officially closed because of the partial government shutdown.
A park ranger secures a road at the entrance to the Mount Rushmore National Memorial on October 1 in Keystone, South Dakota.
A sign is posted in the window of an IRS office in Brooklyn notifying that the office is closed due to the government shutdown on October 1.
A visitor takes a picture of a sign announcing the closure of the Fort Point National Historic Site due to the partial government shutdown on October 1 in San Francisco, California.
A hand-written sign informs visitors to Faneuil Hall, the nation's oldest public meeting hall, that restrooms are closed as a result of the partial government shutdown in Boston, on October 1.
Visitors to Independence National Historical Park are reflected in the window of the closed building housing the Liberty Bell, on October 1 in Philadelphia.
Mark Weekley, superintendent at the National Park Service's Lewis and Clark National Historical Trail, puts up a sign proclaiming the facility closed due to the federal government shutdown, in Omaha, Nebraska, on October 1.
Hot Springs National Park employee Stacy Jackson carries a barricade while closing Arlington Lawn in Hot Springs National Park in Arkansas on October 1.
The Washington Monument is seen behind a chain fence in Washington, on October 1.
A National Park Service ranger finishes putting up a sign indicating all facilities at the Martin Luther King Historic Site are closed to the public in Atlanta, on October 1.
A Capitol police officer walks through the empty Capitol Rotunda, closed to tours during the government shutdown on Capitol Hill in Washington, on October 1.
An employee at the Springfield Armory National Historic Site in Springfield, Massachusetts, puts up a sign on October 1, to notify visitors that the site is closed because of a government shutdown.
A U.S. Park Service police officer stands at the closed Martin Luther King Jr. Memorial on the National Mall in Washington on October 1.
A man looks into the closed Smithsonian National Air and Space Museum in Washington on October 1.
A National Parks Service ranger posts a sign on the doors of the historic Ebenezer Baptist Church in Atlanta on October 1 notifying visitors that the church is closed.
A U.S. park ranger places a closed sign on a barricade in front of the World War II Memorial in Washington on October 1.
Park police and Park Service employees close down the Martin Luther King Jr. Memorial on the National Mall on October 1.
A sign informs visitors that the Suresnes American Cemetery and Memorial, west of Paris, is closed because of the shutdown on October 1.
A man walks past a sign noting the closure at the Cuyahoga Valley National Park in Valley View, Ohio, on October 1.
Members of the U.S. National Park Service close the Lincoln Memorial on the National Mall in Washington on October 1.
A U.S. park ranger posts a closed sign at the Lincoln Memorial on October 1.
A sign alerting visitors that the National Gallery of Art is closed stands outside the building on October 1.
People look at a sign announcing that the Statue of Liberty is closed in New York on October 1.
Fencing around the World War II Memorial prevents people from entering the monument on the National Mall in Washington on October 1.
Signs taped on museum doors alert visitors that the National Museum of American History in Washington is closed on October 1.
A U.S. park service police officer stands guard at the entrance of the closed Lincoln Memorial on October 1.
An employee hangs a sign on the door of the Smithsonian's National Air and Space Museum in Washington on November 14, 1995, marking the start of the government shutdown.
A tourist peers out a ferry window at the Statue of Liberty on November 14, 1995, as a small group of visitors wait on the dock to board the vessel. No passengers were allowed off the boat as both the Statue of Liberty and Ellis Island were closed after federal workers were sent home.
Sen. Pete Domenici, R-New Mexico, chairman of the Senate Budget Committee holds up a chart showing the differences between Republican and Democratic budgets as Speaker of the House Newt Gingrich, left, and Senate Majority Leader Bob Dole stand by during a press conference on Capitol Hill.
A police officer walks through the empty Statuary Hall in the Capitol on November 15, 1995.
The national debt clock in New York is stopped during the government shutdown in November.
President Clinton speaks about the federal budget impasse from the Oval Office on November 16, 1995. The first part of the budget shutdown ended on November 19 when a temporary spending bill was enacted. But Congress failed to come to an agreement on the federal budget, leading to a second shutdown starting December 16.
UPS workers deliver letters to members of Congress on November 28, 1995. The letters were written and sent by members of the Coalition For Change, a nonpartisan organization devoted to balancing the budget.
A speaks to demonstrators at the Capitol Rotunda on December 7, 1995. Evangelical leaders from around the country held a prayer session to call on legislators to treat the poor justly during welfare reform and budget negotiations.
House Appropriations Committee Chairman Rep. Bob Livingston, right, holds a "closed" sign outside the National Gallery of Art in Washington on December 18, 1995.
A security guard informs people that the passport office is closed at the Federal Building in Los Angeles on December 18, 1995.
Rep. John Boehner dumps out coal, which he called a Christmas gift to President Clinton, during a news conference about the federal budget on December 21, 1995.
Karen Bishop chains herself to colleagues during a rally at the Federal Building in San Francisco on January 3, 1996. The workers claimed they were in servitude to the government as hundreds of thousands of federal employees were either furloughed or had to work without pay.
Food service employees at the Veterans Hospital in Miami line up to receive food rations on January 3, 1996. Many federal employees faced financial hardships during the shutdown.
Tourists line up outside the National Gallery of Art in Washington on January 5, 1996. It was one of the few government buildings open during the shutdown thanks to the assistance of private funds.
People trying to apply for visas at the U.S. consulate in Paris on January 5, 1996, are told that the building is closed because of the U.S. budget crisis.
Rep. Thomas Davis III, R-Virginia, attends a rally in Washington on January 5, 1996, urging the end of the government shutdown.
Tourists view Yosemite National Park in California after it re-opened on January 6, 1996. Early that morning, President Clinton signed Republican-crafted legislation to restore jobs and provide retroactive pay to government workers while he and Congress continued negotiating how to balance the federal budget.
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