The Ohio congressman and Freedom Caucus member announced Thursday that he will challenge the expected bid by Majority Leader Kevin McCarthy, who is current speaker Paul Ryan's hand-picked successor.
Allies of President Trump have targeted the deputy attorney general over what they call foot-dragging by the Justice Department in response to requests for information.
Deputy Attorney General Rod Rosenstein told Congress that he knew the president planned to fire James Comey before he wrote a memo that the White House had cited to justify the termination.
After a closed meeting with Rod Rosenstein, senators say the deputy attorney general "answered a lot of questions but declined to answer a lot, as well" about the firing of FBI Director James Comey.
Deputy Attorney General Rod Rosenstein has appointed Robert Mueller to oversee the growing probe into Russian meddling in the 2016 election and possible ties to Trump associates.