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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.
"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.
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."
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.
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."
CNN's Dana Bash and Alan Silverleib contributed to this story.