As the shutdown stretches on, a bloc of moderate House Republicans could be the key to reopening government.

On Wednesday, Speaker John A. Boehner, R-Ohio, held meetings with groups of “pragmatist” lawmakers — as Michael G. Grimm, R-N.Y., described them — who want to pass a policy-rider-free continuing resolution and end the government shutdown as soon as possible.

Grimm said the group was “spitballing some ideas” on how to pass a CR that would fund the entire government, but he indicated that any plan would probably require a number of centrists to join Democrats in voting down a routine procedural motion in an attempt to seize control of the debate and the House floor.

Grimm also expressed support for wrapping negotiations over the debt limit, sequester and the CR into one.

“I do feel we’re moving in the right direction, but for me, it can’t be fast enough,” Grimm said.

It isn’t fast enough for Rep. Peter T. King of New York, who was one of the most vocal House Republicans criticizing the party’s strategy as the government headed to a shutdown.

King wasn’t invited to any of Boehner’s moderate meetings Wednesday, so he held his own.

King said he met in his office with roughly 10 members who support a clean CR, and they discussed “what the strategy would be.”

“Everyone wants a clean CR; some just have different timelines” for action, King said.

But King said ultimately Republicans were going to agree to a clean CR.

“It would probably make it easier if we can show that we defeated Obamacare, even if we didn’t,” King said.

Another moderate Republican, Rep. Mike Simpson of Idaho, summed up the current state of play.

“The real problem is we may have gotten ourselves in a position where we can’t budge on a clean CR and they can’t budge on Obamacare,” Simpson said. “I don’t think closing down the government is a good strategy for us.”

But it does seem to have caught the attention of Democrats and the nation. On Wednesday, Boehner headed to a meeting with the president and other congressional leaders, and lawmakers on both sides were hoping they could begin to forge an agreement.

If those talks don’t bear fruit, centrists could decide to try to hijack the floor. One way to get a clean CR would be by voting down a motion to order the previous question. If GOP centrists joined Democrats to vote down the previous question, Democrats could get control of the floor for an hour and might be able to offer a clean CR.

But when would a group of moderates actually employ that strategy?

“That’s the question,” Grimm said. “I think everyone is trying to give leadership at least the opportunity to have the conversation with the other side.”

Asked when he thought the group of moderate Republicans would start voting to do that, King wasn’t sure.

“I think they should do it now,” he said. “Believe it or not, people don’t always listen to me.”

King said that was precisely what lawmakers in his office were discussing — “when they should do it, how it should be done, what process we should follow.”

He said there were many lawmakers who support a clean CR. He cautioned, however, that many of them may never be in a position to vote for it “because of their districts, they’re afraid, they’re concerned about a primary.”

“One thing I admire about the Ted Cruz Republicans, they don’t care what anyone thinks about them, they just go ahead and do it. I think that’s what we have to start doing,” King said. “So to that extent I’m a Cruz Republican.”

But until centrists start acting with more abandon, Boehner is likely to stick with the current plan — even if they have the numbers right now to force a vote on a clean CR.

It’s no secret that King and Rep. Charlie Dent of Pennsylvania have been leading a contingent of moderates, and as of Wednesday afternoon, there were at least 18 Republicans who had publicly stated they would vote for a clean CR. The list includes King, Dent, Grimm, Simpson, Devin Nunes of California, C.W. Bill Young of Florida, Erik Paulsen of Minnesota, Frank A. LoBiondo of New Jersey, Leonard Lance of New Jersey, Jon Runyan of New Jersey, Lou Barletta of Pennsylvania, Jim Gerlach of Pennsylvania, Michael G. Fitzpatrick of Pennsylvania, Patrick Meehan of Pennsylvania, Frank R. Wolf of Virginia, J. Randy Forbes of Virginia, Rob Wittman of Virginia and Scott Rigell of Virginia.

If they all stick together, the bloc could make Boehner’s life difficult. But Boehner is worried about another group.

Many speculate that Boehner would face a conservative mutiny if he simply brought up a clean CR to be passed by Democrats and a small number of moderate Republicans.

Boehner needs at least 116 Republicans to vote for a CR. That’s half of the Republican Conference’s 232 members, and if Boehner had 116 Republicans to vote for a clean CR, the conservative mutiny would have to stand up to the majority of Republicans in the conference.

But there aren’t 116 Republicans supporting a clean CR — at least not yet and not publicly.

And the moderate group is still waiting to see if Boehner can pull out a deal with Senate Majority Leader Harry Reid and President Barack Obama.

“I think there’s going to be another solution found,” Grimm said.“I know that Speaker Boehner understands the gravity of this situation.”

But ultimately, King said the moderates were going to have fight it out with the far right.

“First of all, I don’t know who they could elect besides John [Boehner]. And sooner or later, we’re going to have to face up to these guys,” King said.

Emma Dumain contributed to this report.