Rep. Tom Price will not challenge Speaker John A. Boehner of Ohio for the gavel, his staff announced Monday, but the fact that he is being floated as a contender by conservatives shows the continued unease and suspicion of the House leadership team.

“Congressman Price is not running for speaker. He is focused on real solutions to get America back on track,” Price spokesman Ryan Murphy said.

The comment comes after a National Review article hinted at a possible Price run if the speaker extends himself too much in a fiscal cliff deal with President Barack Obama. The storyline seems to have come from American Majority, a conservative organization that started a “FireBoehner” Twitter hashtag and is pushing Price, outgoing Republican Conference Chairman Jeb Hensarling of Texas and outgoing Republican Study Committee Chairman Jim Jordan of Ohio as possible alternatives to the existing leadership team.

“The #FireBoehner movement has become a threat: If Speaker Boehner compromises with Obama and they propose a tax-hike together, Speaker Boehner will be finished,” the group’s president and CEO, Ned Ryun, wrote in a Monday blog post on RedState.com.

Aides to Hensarling and Jordan also denied they would run. “The only leadership position Congressman Hensarling plans to hold in the 113th Congress is chairman of the House Financial Services Committee,” said Sarah Rozier, Hensarling’s communications director.

A leading conservative aide dismissed the idea, too, saying there is no talk of a challenge.

Still, this is the latest episode in a long history of conservative dissatisfaction with Boehner, and it could make for an awkward dynamic when the GOP huddles at its annual weekend retreat Jan. 16-18.

Most recently, conservatives have grumbled that Reps. Tim Huelskamp, R-Kan., Justin Amash, R-Mich., and David Schweikert, R-Ariz., were removed from committee assignments because of their ideology — a claim Boehner has denied.

Any challenge would be a long shot, however, as Boehner was already nominated without opposition last month. Rep. Louie Gohmert, R-Texas, tried his hand at a challenge by nominating former Speaker Newt Gingrich, R-Ga., but nobody seconded the nomination.

Price, meanwhile, has options. He narrowly lost a bid for the conference chairmanship, but he could challenge Sen. Saxby Chambliss, R-Ga., one of Boehner’s best friends, in a primary in 2014.