The Texas attorney general, Ken Paxton, in the midst of an attempt to purge Republicans who supported a failed impeachment effort to depose him from power, is picking a fight with Texas’s senior senator, John Cornyn, who is seeking to succeed Senator McConnell as the GOP leader.

In the wake of Mr. McConnell’s announcement on Wednesday that he will step down as the Senate’s GOP leader, Mr. Cornyn declared that he is launching a bid to be the new Republican leader.

“Throughout my time I’ve built a track record of listening to colleagues and seeking consensus, while leading the fight to stop bad policies that are harmful to our nation and the conservative cause,” Mr. Cornyn said in a statement Thursday.

Mr. Cornyn is one of three GOP leadership members who have been loyal to Mr. McConnell and are now expected to vie for the leadership position, with Senators Thune and Barrasso being the others.

Mr. Paxton has publicly voiced his opposition to Mr. Cornyn’s candidacy — a reflection of the campaigning Mr. Paxton has done against other, lower-level Texas Republicans who he would like to replace with Republicans more loyal to him and to President Trump.

“It will be difficult for Cornyn to be an effective leader since he is anti-Trump, anti-gun, and will be focused on his highly competitive primary campaign in 2026,” Mr. Paxton said in a post, possibly alluding to a Senate bid himself.

Mr. Cornyn responded, saying, “Hard to run from prison, Ken,” likely in reference to Mr. Paxton’s securities fraud trial set for April. Mr. Paxton is charged with soliciting investors for a technology company without disclosing that the company was paying him to promote its stock.

This isn’t the first time that Messrs. Paxton and Cornyn have crossed swords or even launched personal attacks against one another.

Earlier this month Mr. Paxton attacked Mr. Cornyn for a late night foreign aid vote in the Senate, saying in a post it is “unbelievable” that Mr. Cornyn “would stay up all night to defend other countries borders, but not America.”

Mr. Cornyn responded, again attacking Mr. Paxton for his legal troubles. “Ken, your criminal defense lawyers are calling to suggest you spend less time pushing Russian propaganda and more time defending longstanding felony charges against you in Houston, as well as ongoing federal grand jury proceedings in San Antonio that will probably result in further criminal charges,” Mr. Cornyn posted.