Two GOP senators are cautioning former President Trump from getting involved in the high-stakes race to replace Senate Minority Leader Mitch McConnell (R-Ky.) when he leaves Republican leadership later this year.

The big picture: Trump — the 2024 GOP presidential frontrunner consolidating his power over Republicans in Congress — has been quietly pushing Sen. Steve Daines (R-Mont.) to run for GOP Senate leader.

What they're saying: Sen. Markwayne Mullin (R-Okla.) said his advice to Trump is to "kind of stay out of the race because it's a lose-lose situation," per an interview on CNN's "State of the Union" Sunday.

• "Whatever leader is there understands that they're gonna have to work with President Trump, too, so it's really not in his best interest to lean in the race at this point," Mullin added.
• "However, if he chooses to do it, it will make a difference," he said.

Sen. Mike Rounds (R-S.D.) said "the former president will have the opportunity to influence a number of my colleagues" when asked Sunday on ABC's "This Week" about Trump's role in the race to replace McConnell.

• "But, we also want to be able to have a good working relationship with him if he becomes the next president of the United States. We've got things we've got to get done," Rounds said.

Zoom in: Both GOP senators echoed their support for Sen. John Thune (R-S.D.), the Senate Minority Whip.

• "I've seen his leadership skills move around, even when Mitch [McConnell] was unable to be at the Capitol because of some illnesses. I saw Thune step up in a very effective manner," Mullin said.
• Similarly, Rounds said he thinks Thune is "the right guy at the right time" and has "great moral character."

Catch up quick: Last week, 82-year-old McConnell, the longest-serving Senate party leader, announced his plans to step down following the 2024 election.

• Thune is one of the "three Johns" — along with fellow GOP Sens. John Barrasso (R-Wyo.) and John Cornyn (R-Texas) — at the center of speculation about who will replace McConnell.

Thune is viewed by many in D.C. circles as the most obvious heir with his current No. 2 position, moderate views and fundraising prowess, Axios' Stef Kight reports.

• Trump endorsed Thune's challenger in 2022, and Thune only endorsed Trump a couple of days before McConnell's announcement that he was stepping down from leadership.

Go deeper: McConnell to step down as Senate GOP leader after 2024 election