In the aftermath of House Majority Leader Kevin McCarthy's decision to drop out of the race, Boehner has said repeatedly he's willing to stay in the speakership until his successor is elected. But if Ryan passes up the job, there will be an all-out dash, with close to a dozen potential candidates scrambling to win the 218 votes necessary to become speaker.
If that occurs, Boehner supporters are worried that it could seriously jeopardize the ability of Congress to reach an accord with the White House on hot-button fiscal issues, potentially prompting the first-ever U.S. default in November and a government shutdown in December. So privately, Boehner's allies now believe that leadership elections should be delayed until later in the year, allowing the Ohio Republican to cut controversial fiscal deals and giving the conference time to coalesce behind a new speaker candidate.
Some are beginning to speak out publicly.
Rep. Tom Cole, R-Oklahoma, told CNN Monday that "there are quite a few" GOP lawmakers who would want Boehner to stay on board.
"We would feel really comfortable -- absent Paul Ryan -- if John Boehner remained until at least we got through those issues, and we had time to conduct a normal kind of (leadership) contest," Cole said. "And we could start with someone new in January or frankly at the beginning of the next term."
One House member, who asked not to be named, also said he planned to make a similar appeal for Boehner to stay on board if Ryan passes up the gavel.
But that move would prompt outrage from the conservative wing of the party that has long sought to ouster Boehner. Rep. Raul Labrador, an Idaho Republican and a leader of the conservative House Freedom Caucus, told CNN Monday that the conference should have a day-long election until a candidate emerges who can unify the conference.
"I do think the establishment in Washington really wants one person to be coronated, and I don't think that should be the process," said Labrador, who insisted that Ryan first lay out a "reform" agenda to win support from the right.
The talk comes as Congress returns from a week-long recess -- after their leadership team was thrown into turmoil following Boehner's sudden decision to resign by month's end and McCarthy's abrupt announcement that he wouldn't seek the speakership. Since then, pressure has been growing on Ryan -- the party's 2012 vice presidential candidate -- to take the speakership, hoping his support on the right could help rebuild a badly divided party.
But Ryan, who is the House Ways and Means Committee chairman, spent the week at home in Janesville, Wisconsin, staying quiet about the job. Ryan remains uncertain about taking the speakership given his skepticism that he'd be able to heal deep divides in the party, he'd have to travel the country with three young children and he'd be unable to dive deeply into his pet policy issues like tax reform, his allies say.
A Ryan spokesman said the Wisconsin Republican "looks forward to listening to and speaking with his colleagues this week."
He'll have a chance to do that Tuesday. The GOP Conference scheduled an unusual meeting Tuesday evening where dinner will be served and staff will be excluded. That meeting comes on top of two all-member meetings set for Wednesday, as the party scrambles to figure out the way forward.
To become speaker, a GOP candidate must win 125 votes and later be elected with 218 on the House floor. It's uncertain which candidate could get to 218, but many are already throwing their hats in the ring if Ryan declines to run.
The possible candidates include Reps. Bill Flores of Texas, Mike Pompeo of Kansas, Lynn Westmoreland of Georgia, Jason Chaffetz of Utah, John Kline of Minnesota and Marsha Blackburn of Tennesee. Rep. Daniel Webster of Florida, who is running for the job, has not said if he'd drop out of the race if Ryan won the party's nomination.
"We're still trying to figure out who is the right person," Pompeo said in an interview Monday. "If it turns out it's me, and I conclude I can do it well, then I'll get in. There are still lots of conversations taking place."
With so many candidates running -- at a time of major fiscal calamities awaiting the next speaker -- Republicans think it makes for Boehner to remain as speaker and maintain some continuity. No other potential candidate, Cole said, has the same "natural stature that Paul does," meaning that Boehner should give time for leadership races to play out before stepping aside.
If Boehner does stay on board, Cole noted it would be "one of the great ironies of all time," given that conservatives cheered his resignation plans but ultimately may be stuck with him because of their refusal to back a consensus candidate.
Boehner has previously thought about resigning on his 66th birthday on November 17, but what he decides to do remains to be seen. His office said that the speaker still believes the party will unite behind a candidate before he plans to resign October 30.
"Speaker Boehner is confident that a new speaker will be elected by the end of the month," said Emily Schillinger, a Boehner spokeswoman.