Rep. Trey Gowdy’s bid to be the next chairman of the Oversight and Government Reform Committee keeps getting easier as a key potential rival says he won’t run and predicted the South Carolina Republican is likely to wield the gavel, even as the deadline to make a bid draws near.

Any serious bid to challenge Gowdy will need to get under way soon, as Speaker Paul D. Ryan is giving members interested in the post until June 1 to let the House Republican Steering Committee know, according to a Ryan spokeswoman. The Steering Committee is on track to vote on the next chairman the week following the Memorial Day recess.

One member who posed a significant threat to Gowdy’s bid has decided he won’t go for the post. “You guys know how this works,” Rep. Jim Jordan of Ohio told reporters Monday. “The establishment’s not going to put the anti-establishment guy in charge of the committee whose job it is to go after the establishment. It’s just not going to happen.”

Jordan, a founder of the House Freedom Caucus, said his and his group’s conflicts with leadership would likely sink any chance he had. 

The Steering Committee, dominated as it is by leadership and its surrogates, Jordan said, is “just not the most favorable audience for me.” Jordan thinks Gowdy is the frontrunner.

“Trey’s probably going to be the  guy,” he said. “Trey’s a good guy.”

Republican California Rep. Darrell Issa, a former chairman of the committee who can technically take the gavel again, also predicted Gowdy would prevail.

“I would assume as we look down the dais that he would be on everyone’s short list,” Issa said.

When asked if he would be interested in running, Issa didn’t rule it out entirely saying he was “never without interest in anything” but that becoming a committee chairman for a second time was unprecedented.

After the Steering Committee votes after the Memorial Day recess, the full Republican Conference will then vote. The member chosen will assume the chairmanship when it is read on the House floor, which leadership intends to do before the current chairman, Utah Rep. Jason Chaffetz, leaves Congress on June 30.

Gowdy gained national attention as the chairman of the special House committee investigating the 2012 Benghazi incident. He at one point led the questioning of Hillary Clinton in a marathon panel hearing during the 2016 presidential campaign. He also was criticized by Democrats and some staffers for a lack of objectivity in his proceedings. 

A spokesman for the South Carolina Republican would only say Gowdy was considering a run and was speaking with members about what they were looking for in a chairman of a committee charged with probing the executive branch.

As for Chaffetz, he’s opting to let others decide who to pick for his replacement.

“Staying out of it,” he replied when asked if he planned to make an endorsement.

Contact Rahman at remarahman@cqrollcall.com or follow her on Twitter at @remawriter