Three other Republicans in the race out-raised Mr. Greitens: Representative Vicky Hartzler, Attorney General Eric Schmitt and Mark McCloskey, the man best known for waving his gun outside his St. Louis home as protesters marched last year. Some national Republican strategists are worried that if Mr. Greitens survives a crowded primary, he could prove toxic even in a heavily Republican state.
Mr. Scott has pledged to remain neutral in party primaries, but Senator Mitch McConnell of Kentucky, the Republican leader, has long preferred promoting candidates he believes can win in November.
“The only thing I care about is electability,” Mr. McConnell told Politico this year. With Mr. Scott on the sidelines, a McConnell-aligned super PAC, the Senate Leadership Fund, is expected to do most of the intervening.
Mr. Trump, who is often at cross-purposes with Mr. McConnell, has appeared especially engaged in the Arizona and Georgia races, largely because of his own narrow losses there. He has publicly urged the former football player Herschel Walker to run in Georgia — Mr. Walker has not committed to a campaign — and attacked the Republican governor of Arizona, Doug Ducey, even after Mr. Ducey has said he is not running for Senate. Some Republican operatives continue to hope to tug Mr. Ducey into the race.
Mr. Trump delivered one early Senate endorsement in North Carolina, to Representative Ted Budd, who raised $953,000, which is less than the $1.25 million that former Gov. Pat McCrory pulled in. Some Republicans see Mr. McCrory as the stronger potential nominee because of his track record of winning statewide.
In Alaska, Kelly Tshibaka is running as a pro-Trump primary challenger to Senator Lisa Murkowski, who voted to convict Mr. Trump after his second impeachment. Ms. Murkowski, who has not formally said if she is running again, raised more than double Ms. Tshibaka in the most recent quarter, $1.15 million to $544,000.
In Alabama, Mr. Trump gave another early endorsement to Representative Mo Brooks and recently attacked one of his rivals, Katie Britt, who is the former chief of staff of the retiring incumbent, Senator Richard Shelby. Ms. Britt entered the race in June, but she out-raised Mr. Brooks, $2.2 million to $824,000. A third candidate, Lynda Blanchard, is a former Trump-appointed ambassador who has lent her campaign $5 million.