That has created tensions between Senator Mitch McConnell of Kentucky, the majority leader who strongly backs Mr. Strange, and the West Wing, where some sympathies lie with Mr. Brooks. What Mr. Trump does will beam a message to his supporters well beyond Alabama about which Republican faction he favors: the pragmatic establishment or uncompromising hard-liners.
And in Alabama, Mr. Trump’s preference matters. He may be saddled with national disapproval ratings that no modern president has seen this early in an administration, but among Alabama conservatives, the president ranks up there with college football, air-conditioning and pork shoulder.
“Inside a Republican primary, the only person who may potentially be more popular is Nick Saban,” said Clay Ryan, the chief lobbyist for the University of Alabama system, referring to the all-but-deified head football coach of the Crimson Tide.
Allies of Mr. Strange are aggressively lobbying Mr. Trump to intercede, circulating Mr. Brooks’s harsh campaign comments about the president’s bouts of alleged marital infidelity. The president is generally aware of Mr. Brooks’s criticism, lobbed during the 2016 primary when Mr. Brooks backed Senator Ted Cruz of Texas, according to multiple Republicans inside and beyond the administration. But he has kept his counsel.
Mr. Brooks said he had no regrets about his tart criticism.
“When I endorse a candidate, as I endorsed Ted Cruz, I’m a fighter and I’m all in,” Mr. Brooks said.
Where Mr. Trump ends up is anybody’s guess. He has sent positive signals to Mr. Strange, who has spoken to the president directly and accepted every opportunity to go on Fox News and sing Mr. Trump’s praises. But so far the president has declined to offer an endorsement.
Presidents almost always support congressional incumbents of their own party. But Mr. Strange was not elected in his own right and his appointment carries the odor of the most recent unpleasantness in Alabama’s political scene: He was handed the Senate seat while serving as state attorney general by Robert Bentley, then the governor, who not long after resigned in the face of a sex and abuse of power scandal.
Mr. Strange’s appointment was made under “an ethical cloud,” Mr. Brooks said.
The White House’s reluctance stems not so much from the circumstances of Mr. Strange’s ascent but from the forces that are lining up behind Mr. Brooks. The Huntsville-area congressman has collected endorsements from ardent Trump allies in conservative news media circles like Sean Hannity and Laura Ingraham, as well as from fellow members of the far-right House Freedom Caucus, who have urged the White House to give Mr. Brooks a clear berth in the race, according to Republicans briefed on the conversations.
Mr. Brooks was dismissive of Mr. McConnell’s efforts to shape the race, and said that if he won the Senate seat, he might not support Mr. McConnell as majority leader.
The race also has the potential to grow even messier than a head-to-head between Mr. Brooks and Mr. Strange. A third contender, Roy S. Moore, the former chief justice of the Alabama Supreme Court who made his name by refusing to remove the Ten Commandments from the court’s grounds, could upset the contest with the passionate support of religious conservatives.
Mr. Trump’s ardent neutrality has spilled over to the Republican National Committee, which is still effectively run by its former chairman, White House Chief of Staff Reince Priebus. Mr. McConnell — who views Mr. Strange as a reliable party-line vote and has led the push for his new colleague — has repeatedly met resistance from the White House in his effort to shore up Mr. Strange.
The Republican National Committee initially did not respond to a request from the Senate Republican campaign arm for funding to back Mr. Strange. That prompted Mr. McConnell’s top political lieutenant, Josh Holmes, and then Mr. McConnell himself to contact the White House.
But the committee still did not authorize the funding. Stunned by the rebuff, Mr. McConnell contacted Mr. Trump directly last month, insisting that the White House had to be on the same page as the Senate leadership, people briefed on the conversation said. Mr. Trump responded by calling Mr. Strange to smooth things over, assuring the new senator that he had a friend in the White House.
Even then, the Republican National Committee did not sign off on the funding until Mr. McConnell’s intervention with Mr. Priebus was publicly reported by The New York Times last month. But while the party money has finally been addressed, aides to Mr. Trump — including Mr. Priebus — believe that it would be a needless risk for Mr. Trump to take sides in a contest where all the candidates describe themselves as strongly pro-Trump.
“There’s a little bit of a turf war because McConnell thinks he’s head of the party and Trump thinks he’s head of the party,” said David Ferguson, an Alabama Republican strategist. “But if Trump really wants to take more than a figurehead role in the party, he’s got to flex his muscles in a Republican primary somewhere, somehow.”
Mr. Strange could use the muscle. Mr. Brooks received a surge of publicity after surviving the shooting on members of Congress last month at a Virginia baseball field, and his profile has grown beyond his congressional district. His rise is alarming Mr. Strange.
The senator sent a mailer to Alabama Republicans last week highlighting Mr. Brooks’s criticism of Mr. Trump. That attack represents only the initial salvo in the assault on Mr. Brooks, which a well-financed “super PAC” run by Mr. McConnell’s allies is expected to lead before the first balloting.
On Mr. McConnell’s instructions, the Republican Senate campaign committee also has threatened to deny business to any consultants who work against Mr. Strange.
That has effectively frozen out other candidates from hiring advisers and fund-raising, said Randy Brinson, an evangelical activist in Alabama who is also running for Senate. Mr. Brinson said he had secured a nationally known pollster, Whit Ayres, to steer his bid, only to get an apologetic email from Mr. Ayres after he backed out.
Still, it is Mr. Brooks whose candidacy is seen by the party establishment as the most significant threat. Mr. Strange’s goal is to weaken Mr. Brooks so that if the race goes to a September runoff election, it is more likely to be between Mr. Strange and Mr. Moore. Mr. Strange’s allies believe Mr. Moore would be far easier for Mr. Strange to defeat in a head-to-head race.
But before then, the 6-foot-9 lawmaker known as “Big Luther” may ultimately not get the Air Force One visit to Birmingham or the presidential Twitter post that he covets. Quin Hillyer, a conservative commentator based in Alabama, saw no upside for Mr. Trump to back Mr. Strange.“He would tick off a rather prominent member of the Freedom Caucus in Mo Brooks, and Mo Brooks’s supporters,” Mr. Hillyer said. “He would tick off all the Roy Moore supporters, who also tend to be strong Trump supporters. I don’t think there’s anything in it for Trump.”