Three other Republicans in the running outperformed Mr. Greitens: Rep. Vicky Hartzler, Attorney General Eric Schmitt, and Mark McCloskey, best known for waving his gun outside his St. Louis home when protesters marched last year. Some national Republican strategists fear that if Mr. Greitens survives a crowded primary, he could prove toxic even in a heavily Republican state.

Scott has promised to remain neutral in the party’s primary election, but Kentucky Senator Mitch McConnell, the Republican leader, has long preferred promoting candidates he believes can win in November.

“The only thing that matters to me is eligibility,” McConnell told Politico this year. With Mr. Scott on the sidelines, a McConnell-sponsored super-PAC, the Senate Leadership Fund, is expected to handle most of the interventions.

Mr. Trump, who often argues with Mr. McConnell, has been particularly involved in the races in Arizona and Georgia, largely because of his own narrow losses there. He has publicly urged former soccer player Herschel Walker to run in Georgia – Mr Walker has not signed up to a campaign – and attacked Arizona Republican Governor Doug Ducey, even after Mr Ducey said he was not running for the Senate is running. Some Republican agents continue to hope to pull Mr. Ducey into the race.

Mr. Trump gave early Senate approval to North Carolina MP Ted Budd, who raised $ 953,000, which is less than the $ 1.25 million withdrawn from former Governor Pat McCrory. Some Republicans see Mr. McCrory as the stronger potential candidate because of his track record in winning nationwide.

In Alaska, Kelly Tshibaka is running as a pro-Trump challenger for Senator Lisa Murkowski, who voted for Trump’s conviction after his second impeachment. Ms. Murkowski, who has not officially said whether she will run again, more than doubled Ms. Tshibaka in the most recent quarter, from $ 1.15 million to $ 544,000.

In Alabama, Trump gave MP Mo Brooks another early endorsement and recently attacked one of his rivals, Katie Britt, the former chief of staff for retired incumbent Richard Shelby. Ms. Britt entered the race in June, but she raised Mr. Brooks by $ 2.2 million to $ 824,000. A third candidate, Lynda Blanchard, is a former Trump-appointed ambassador who loaned $ 5 million to her campaign.