Republican Conference Chairman Jeb Hensarling is a lock to lead the Financial Services Committee next year.
Rep. Ed Royce (Calif.), a vice chairman of the National Republican Congressional Committee, is the top candidate for the post, GOP aides said.
"Royce is an experienced and proven leader on the full range of issues facing the House Foreign Affairs Committee. He looks forward to making his case to his colleagues after the election," spokeswoman Audra McGeorge said.
Rep. Chris Smith (N.J.) is also seeking the position, but his party loyalty has previously been called into question. He was stripped of the chairmanship of the Veterans' Affairs Committee in the 109th Congress, when he bucked leadership and called for increases in veterans' health care spending.
"That's not forgotten," a GOP leadership aide said. "Would it be an issue that happens again? That's something you'd certainly see leadership asking."
Judiciary Chairman Lamar Smith (Texas), who is term-limited in his current job, is looking to head the Science, Space and Technology Committee, a panel he said was his first choice when he entered Congress because of his background studying astronomy and physics in college.
"I am proud of our work on the Judiciary Committee. If given the opportunity, I hope to be just as active of a Chairman of the Science & Technology Committee," he said in a statement. "If America is going to remain competitive in today's global economy, we need to remain innovative and focused on exploring science and expanding new technologies."
And though GOP aides said Smith is the frontrunner, he has competition, and the race is getting heated.
Rep. Jim Sensenbrenner (Wis.) is looking to return to the top of the committee he chaired from 1997 to 2001.
Sensenbrenner also chaired the Judiciary Committee from 2001 to 2007, and he said Smith is only the lead candidate because "he's been tooting his horn more."
"The point is I've been a committee chairman for 10 years," Sensenbrenner said. "The Science Committee was on the map for the six years between 1994 and 2000. ... It kind of has slipped since then in terms of viewing it as an important committee and I want to bring it back."
Rep. Dana Rohrabacher (Calif.) is also trying for the gavel. GOP aides said he is viewed as a loose cannon, however, which could diminish his chances. But Rohrabacher said the Steering Committee should choose someone with creative ideas, rather than a "go-along, get-along type of personality."
"It's only fair to give someone else who hasn't had a chance at being chairman a shot at it," he said.
As Roll Call previously reported, the Steering Committee will have its work cut out in choosing the next head of the Budget Committee should chairman Paul Ryan (Wis.) become vice president or choose not to return to the committee.
Budget Vice Chairman Scott Garrett (N.J.) and Rep. John Campbell (Calif.) have already announced that they intend to go for the gavel, and aides point to Reps. Jason Chaffetz (Utah) and Tom Price (Ga.) as contenders as well, if they want the job.
Then there are the Speaker-appointed chairmanships that are likely to turn over: Rules, Ethics and House Administration.
Speaker John Boehner (Ohio) is considering Natural Resources Chairman Doc Hastings (Wash.) and NRCC Chairman Pete Sessions (Texas) for the Rules Committee job.
Rep. Eric Swalwell, D-Calif., walks on Broadway after a Future Forum with young entrepreneurs in the Flatiron District of New York City, April 16, 2015. Reps. Steve Israel, D-N.Y., Seth Moulton, D-Mass., and Grace Meng, D-N.Y., also attended.