Speculation is growing on Capitol Hill that 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 may need to relinquish his Speaker’s gavel soon, though few Republicans are publicly calling on him to resign.

More rank-and-file Republicans predicted Friday that the Wisconsin Republican probably won’t be able to hang on to the Speaker’s job for the rest of the year, despite Ryan’s insistence a day earlier that he would stick around until January and that no one in the GOP could raise more money ahead of the crucial midterm elections later this year.

“I think there's a lot of goodwill for Paul Ryan, but I don't know if there's so much goodwill that they'll let him stay as Speaker,” said longtime Rep. Joe Barton Joe Linus BartonBottom line Ex-Tea Party lawmakers turn heads on K Street Longtime GOP aide to launch lobbying shop MORE (R-Texas), the former Energy and Commerce Committee chairman who is retiring this year. “I think nobody would have a problem if he resigned the speakership and stayed in Congress to fulfill a commitment to his constituents in Wisconsin."

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“I just think the pressure is going to build for him to step aside as Speaker and then let the conference pick a new leadership team,” Barton added.

So far, few Republicans have been willing to go on the record and call for Ryan to step down, a move that would trigger new leadership elections now rather than after the Nov. 6 elections.

But conservative Rep. Tom Graves John (Tom) Thomas GravesRep. Tom Graves announces early retirement Democrat in race against Marjorie Taylor Greene drops out McEnany: Trump 'hasn't done deep dive' on anti-Muslim views of Loomer, Greene MORE (R-Ga.), an ally of House 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 publicly pushed for a new leadership team to be installed now following Ryan’s announcement this week that he’s not running for reelection this fall.

Graves met with Ryan on Thursday about the matter, The Hill confirmed. “It was a positive conversation. Rep. Graves is confident Speaker Ryan will do what’s best for the conference,” said a Graves aide.

And centrist Rep. Adam Kinzinger Adam Daniel KinzingerFox News reporter defends confirming Atlantic piece despite Trump backlash: 'I feel very confident' GOP lawmaker defends Fox reporter after Trump calls for her firing Lindsey Graham: 'QAnon is bats--- crazy' MORE (R-Ill.) told the Chicago Sun-Times it’d be better to hold those leadership elections “sooner than later.”

“It would be nice to kind of get this stuff done, because what you don’t want is about an eight-month process where people are running for Speaker because it’s very tough,” Kinzinger told the newspaper.

On Friday, a GOP lawmaker closely aligned with leadership told The Hill he, too, would like to see the succession question resolved now.

“There’s a strong sense of wanting to give Paul his breathing room and a chance to talk about his legacy,” said the GOP lawmaker. “But there is a growing sentiment in the conference that the election should be held sooner so that we don’t have a lame-duck Speaker heading into the midterms.”

It’s unclear, however, how many Republicans feel the same way. Speaking to reporters on Thursday, Ryan swatted down questions of whether having an internal, seven-month-long race for Speaker would distract from GOP efforts to hold the House majority. And Ryan said that after holding numerous conversations with fellow Republicans, he had overwhelming support to remain Speaker until January.

“It makes no sense to take the biggest fundraiser off the field. And I think almost all of our members see it that way as well,” Ryan said.

Ryan’s office did not respond to a request for comment for this story.

Among the many GOP lawmakers who’ve said Ryan should finish out his two-year term as Speaker were Reps. Frank LoBiondo Frank Alo LoBiondoVan Drew wins GOP primary in New Jersey Amy Kennedy wins NJ primary to face GOP's Van Drew Stimulus price tag of .2T falls way short, some experts say MORE (R-N.J.), Leonard Lance Leonard LanceThomas Kean wins GOP primary to take on Rep. Tom Malinowski Gun debate to shape 2020 races GOP fears Trump backlash in suburbs MORE (R-N.J.), Pete King Peter (Pete) KingTrump holds private funeral service for brother Robert Trump at White House Cheney clashes with Trump Coronavirus Report: The Hill's Steve Clemons interviews Rep. Sean Patrick Maloney MORE (R-N.Y.), Walter Jones Walter Beaman JonesExperts warn Georgia's new electronic voting machines vulnerable to potential intrusions, malfunctions Georgia restores 22,000 voter registrations after purge Stacey Abrams group files emergency motion to stop Georgia voting roll purge MORE (R-N.C.), Mo Brooks Morris (Mo) Jackson BrooksOvernight Defense: Trump hosts Israel, UAE, Bahrain for historic signing l Air Force reveals it secretly built and flew new fighter jet l Coronavirus creates delay in Pentagon research for alternative to 'forever chemicals' OVERNIGHT ENERGY: House Democrats tee up vote on climate-focused energy bill next week | EPA reappoints controversial leader to air quality advisory committee | Coronavirus creates delay in Pentagon research for alternative to 'forever chemicals' Coronavirus creates delay in Pentagon research for alternative to 'forever chemicals,' official says MORE (R-Ala.), Mark Sanford Mark SanfordOn The Money: Business world braces for blue sweep | Federal Reserve chief to outline plans for inflation, economy | Meadows 'not optimistic' about stalemate on coronavirus deal Trump critic Sanford forms anti-debt advocacy group Republicans officially renominate Trump for president MORE (R-S.C.), Bill Flores William (Bill) Hose FloresPatient Protection Pledge offers price transparency Texas GOP lawmaker calls for 'carbon neutral' but 'energy dominant' future OVERNIGHT ENERGY: Biden calls climate change one of America's four major crises | National parks chief says coronavirus staff shortages shouldn't prevent access | Trump hits California officials over wildfires MORE (R-Texas) and Ryan Costello Ryan Anthony CostellloTrump struggles to stay on script, frustrating GOP again Bottom line Former GOP Rep. Costello launches lobbying shop MORE (R-Pa.).

“I believe the Speaker is best-positioned to lead our conference for the rest of the session, policy-wise, and as the single best fundraiser in the history of the House Republican Conference,” Costello told The Hill. “A leadership contest in the interim is a distraction and could lead to unnecessary divisions.”

The race to replace Ryan had already been shaping up as a contest between McCarthy, Ryan’s top deputy, and Majority Whip Steve Scalise Stephen (Steve) Joseph ScaliseHouse GOP slated to unveil agenda ahead of election House panel details 'serious' concerns around Florida, Georgia, Texas, Wisconsin elections Scalise hit with ethics complaint over doctored Barkan video MORE (R-La.), the No. 3 GOP leader, though neither has publicly declared a bid for Speaker.

Friday added a new wrinkle to the Speaker sweepstakes. Rep. Jim Jordan James (Jim) Daniel JordanSunday shows preview: Justice Ginsburg dies, sparking partisan battle over vacancy before election House passes resolution condemning anti-Asian discrimination relating to coronavirus Republicans call for Judiciary hearing into unrest in cities run by Democrats MORE (R-Ohio), the former chairman of the far-right House Freedom Caucus, said that he was mulling over a bid for the top leadership post.

Rep. Thomas Massie Thomas Harold MassieGOP lawmaker praises Kyle Rittenhouse's 'restraint' for not emptying magazine during shooting Rep. Dan Meuser tests positive for COVID-19 Liz Cheney wins Wyoming GOP primary in reelection bid MORE (R-Ky.), a Jordan ally who is not a Freedom Caucus member, said there’s just no way impatient Republicans can wait eight months to figure out who should be the next leader of the 237-member GOP conference.

“This just seems like such an improbable proposition, that we would have a lame-duck Speaker for eight months,” Massie said. “I don’t know if the palace can sustain that much intrigue.”

“It seems improbable that you could have this kind of vacuum.”

Melanie Zanona and Juliegrace Brufke contributed.