House Republicans are heaping pressure on Rep. Paul Ryan Paul Davis RyanKenosha will be a good bellwether in 2020 At indoor rally, Pence says election runs through Wisconsin Juan Williams: Breaking down the debates MORE (R-Wis.) to become their next Speaker after the strong favorite for the position stunned Washington by exiting the race.

Now that Majority Leader Kevin McCarthy Kevin Owen McCarthyMcCarthy's Democratic challenger to launch first TV ad highlighting Air Force service as single mother Trump asked Chamber of Commerce to reconsider Democratic endorsements: report The Hill's Morning Report - Sponsored by The Air Line Pilots Association - White House moves closer to Pelosi on virus relief bill MORE (R-Calif.) has bowed out, a growing number of House Republicans are arguing that the Ways and Means Committee chairman is the only person capable of uniting a GOP conference that has been badly fractured for years.

“Paul Ryan is the only eligible candidate,” insisted Rep. Devin Nunes Devin Gerald NunesSunday shows preview: Justice Ginsburg dies, sparking partisan battle over vacancy before election Sunday shows preview: With less than two months to go, race for the White House heats up Sunday shows preview: Republicans gear up for national convention, USPS debate continues in Washington MORE (R-Calif.), a frequent critic of the House’s conservative wing who had backed McCarthy for Speaker.

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The push to draft Ryan is coming from the top. McCarthy told National Review that he wanted Ryan to run the House, and The Washington Post has reported that Speaker John Boehner John Andrew BoehnerLongtime House parliamentarian to step down Five things we learned from this year's primaries Bad blood between Pelosi, Meadows complicates coronavirus talks MORE (R-Ohio) has called on the Wisconsin Republican to be his replacement.

Additional GOP heavyweights, such as Rep. Trey Gowdy (R-S.C.), who some wanted to see make a run for the Speakership, are also backing a Ryan bid. Thursday afternoon, Ryan was seen on the House floor chatting with Rep. Mick Mulvaney (R-S.C.), a vocal conservative, and House Financial Services Committee Chairman Jeb Hensarling (R-Texas), a conservative favorite who himself has declined leadership runs in the past.

Other leadership allies, as well as Nunes, were among the first to single out Ryan as the best candidate in the wake of McCarthy’s announcement.

“He’s just the first obvious name and it would be terrific,” said Rep. Tom Cole (R-Okla.). “He has national stature having been a vice presidential nominee, and he’s a conservative icon.”

There’s only a couple of problems with that idea.

First, Ryan has steadfastly insisted he has no interest in the Speaker’s job. He made that clear within minutes of Boehner announcing his resignation last month, and he has not shifted from that position since.

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On top of that, it’s far from certain that the House Freedom Caucus, the group of roughly 40 conservatives who opposed Boehner and McCarthy, would fall in line behind a Ryan candidacy.

“While I am grateful for the encouragement I’ve received, I will not be a candidate,” Ryan said in a statement shortly after McCarthy’s withdrawal announcement Thursday.

A spokesman for Ryan said later in the day that the Wisconsin Republican was still a “hard no,” after GOP lawmakers continued to urge him to run.

Ryan has long called heading the powerful Ways and Means Committee his dream job, giving him the opportunity to overhaul the tax code for the first time in a generation, revamp the country’s poverty programs and deal with increasing entitlement spending.

But the 45-year-old father of three also said after Boehner announced his resignation that the Speaker position was best reserved for “an empty nester.” During his tenure, Boehner has traveled almost constantly, and across prodigious distances, to fundraise for House Republicans.

Taking one of the most difficult jobs in Washington would complicate any ambitions that Ryan, the 2012 GOP nominee for vice president, has for winning the White House one day, given how poorly voters have historically thought of congressional leaders.

Those pushing Ryan acknowledge those challenges, but are reaching the point where they believe the party, and even the country, needs him in that role.

“The challenge obviously is in terms of his thinking about what he wants to do personally, and his family, and the demands this puts on him,” said Cole. “But boy, this is certainly a time where we need him very badly.”

Nunes floated Ryan as the only other option immediately after McCarthy dropped out, even before he had heard that Ryan had put out a statement saying he wasn’t a candidate. But he vowed to press him on the job.

“Already?” Nunes said after being told of Ryan’s statement. “Well, I haven’t talked to him yet, so we’ll see.”

One of the issues for conservatives would be that Ryan, if he ran, would be seen as the hand-picked successor to Boehner and alternative to McCarthy.

“‘Hand-picked by leadership’ doesn’t get much support around here,” said Rep. Tim Huelskamp (R-Kan.). “Two Speakers gone in two weeks.”

Giving the Speaker’s gavel to a high-wattage Republican such as Ryan would also run counter to what members of the House Freedom Caucus have said they want in a leader.

House conservatives, tired of Boehner constantly cutting deals with the Senate and the White House on government funding and other issues, say they want to give more power back to the committees and to the rank-and-file.

“I don’t think he’s the only one,” said Rep. Mark Meadows (R-N.C.), referring to Ryan. “It really gets down to policies, and rules, and procedures and how we change the way we do business here.

“This isn’t a student class election, where you say, ‘This is the person everyone can unite behind,’” added Meadows, who started the procedural motion that paved the way for Boehner to resign.

Those sorts of comments also made it clear that the House conservatives, who had endorsed Rep. Daniel Webster (R-Fla.) for Speaker, are relishing the sway they hold over who the next Speaker will be.

“We’ll probably start another interview process up. They’ll have to go through it,” Huelskamp said. “I don’t think there’s anybody that’s going to stand up and say, ‘I get it. I don’t have to make any promises. No commitments. No changes.’”