However, Thune and Cornyn share an attribute that could complicate their path to leadership: Both have criticized former President Donald Trump and have openly hoped that another candidate would win the GOP presidential nomination in 2024. Thune has endorsed fellow Senator Tim Scott, and there’s no love lost between the mild-mannered South Dakotan and the bombastic former president. Trump called for Thune to be primaried in 2022, and did not endorse him in his reelection bid. Thune did not face a serious challenge in the primary and won the general election handily.

Barrasso, Thune’s colleague from neighboring Wyoming, has vocally supported Trump. Barrasso, who chairs the Senate Republican Conference, is the “strongest on being able to connect to the Trump world,” said a former senior adviser to a top Republican senator. The contest to succeed McConnell could be influenced by external factors—if, say, Trump opposes Thune or Cornyn as majority leader, it could prompt senators on the party’s right flank to back Barrasso.

“I think that Barrasso is the interesting third candidate here,” the former senior adviser said. “Some of those more conservative members of the Senate could step forward and say, ‘Nope, we want somebody who voices more of our positions.’”