House Republicans could support Paul Ryan for speaker. But will he run?

If rebel conservatives and establishment Republicans in the House can agree on anything, it's likely supporting Paul Ryan for speaker.

Across the Sunday talk shows, members of the Freedom Caucus that helped drive Speaker John Boehner toward retirement said they could swing behind Ryan in the wide-open race for a successor. That's what Boehner and his allies want, too. But the fragile, theoretical consensus falls apart over just how the speaker should wield power.

Freedom Caucus members want major changes in how the House works, including shifting some power from their speaker to committee chairmen. The ideal speaker, these conservatives say, would answer to Republicans, even if that leads to such consequences as government shutdowns. Others see the speaker as responsible, ultimately, for governing – the minimum, senior Republicans say, required to maintain GOP majorities in Congress and win the presidency in the 2016 elections.

The debate tangles the Constitution with obscure House rules and palace intrigue. But the answer could help Ryan decide whether to steer a caucus that many say can't be led. The path forward seemed to rest on Ryan's answer, which on Sunday remained, "no thanks."

"Paul Ryan is a good man. He's a great communicator, the kind of messenger I think our party needs," Freedom Caucus Chairman Jim Jordan, R-Ohio, said on "Fox News Sunday." ''And certainly, if he gets in the race, I think our group would look favorably on him."

The prospect of Ryan as speaker arose after a rollicking two weeks on Capitol Hill, in which Boehner announced his resignation and his heir apparent, Majority Leader Kevin McCarthy, abruptly withdrew his candidacy. Freedom Caucus members had presented Mr. McCarthy with a list of demands he and top aides concluded could be next to impossible to meet, including a seat at the leadership table, a role in selecting committee chairs and members, and more votes on their legislative proposals.

A leadership crisis resulted, with Ryan the only person widely deemed acceptable to fill it – under certain conditions. He has said repeatedly he doesn't want the job, but is said to be thinking about it.

One Republican close to Ryan said that the only scenario in which Ryan might end up as speaker is if he were to be selected by unanimous acclamation, not subject to bargaining with the Freedom Caucus. This Republican demanded anonymity to discuss private considerations.

Rep. Raul Labrador of Idaho agreed that it's possible that he and other members of the House Freedom Caucus would support Ryan or others. But first, Labrador said, any candidate needs to talk to the caucus and address the concern that the speaker should more effectively push the Republicans' agenda before making deals.

"It's not about the who, it's about the what," Labrador told CNN's "State of the Union" on Sunday. "What are we going to do in the House to change the culture? What are we going to do so we can get 247 Republicans together on the same page?"

Rep. Mick Mulvaney, R-S.C., said he, too, would consider voting for Ryan. "First, he's got to make a decision to run," Mulaney said. "And then I think he's got to convince me and some other folks that if he were in charge, that the place would be different."

Even Rep. Jason Chaffetz, R-Utah, who launched a bid for speaker late last week, said he could be open to the prospect of Ryan as speaker.

"We allow these good bills and ideas to percolate from the bottom up, rather than the top down-driven process where the speaker is telling the body what to do," Chaffetz, the chairman of the House Oversight and Government Reform Committee, said on ABC's "This Week."

"I think the speaker works for the conference, the House Republicans," he said.

Members of the GOP establishment remained squarely behind Ryan. One, Rep. Tom Cole of Oklahoma, tried to downplay the differences between factions and appealed for calm.

"Our disagreements tend to be tactical, not theological. We actually believe in the same thing," Cole said on CNN. "It's not like we're going to be without a speaker."

Associated Press writers Anne Flaherty and Erica Werner contributed to this report.