If Mr. McCarthy does have a plan, he has not shared it with members of his leadership team, whom he has cut out of his deliberations about the speakership race in what some regard as a display of paranoia. Instead, he has been spotted in recent days around the Capitol and the Republican National Committee headquarters nearby with Jeff Miller, a Republican lobbyist who is among his closest confidants.
It was not clear whether Mr. McCarthy enlisted Mr. Trump to help his campaign, or if Mr. Trump was simply working on his own. The former president has spoken with Eli Crane, an incoming Republican congressman from Arizona, and Representative Ralph Norman, Republican of South Carolina, among others. Mr. Crane and Mr. Norman were part of a group of seven current and incoming Republican lawmakers who signed a letter with a list of concessions they are demanding from their leaders in the next Congress, including making it easier to force a vote to remove the speaker — something that Mr. McCarthy has so far resisted.
Mr. Norman, who has described himself as a “hard no” against Mr. McCarthy, declined to discuss his call with Mr. Trump, describing it as a “private conversation.” He said he was still undecided about whom he would support for speaker. Mr. Crane did not respond to requests for comment.
When Nancy Pelosi in 2018 found herself about a dozen votes short of what she would need to secure the speaker’s gavel, she quietly picked off defectors, methodically cutting deals to capture exactly enough support to prevail. Ms. Pelosi, renowned for her ability to arm-twist and coax, won seven votes by agreeing to limit her tenure, picked up another eight by promising to implement rules aimed at fostering more bipartisan legislating, and won over her sole would-be challenger by creating a subcommittee chairmanship for her.
For Mr. McCarthy, there are likely fewer deals to be made.
The California Republican has already made a series of pledges in an effort to appease the right flank of his party. He traveled to the southern border and called on Alejandro N. Mayorkas, the homeland security secretary, to resign or face potential impeachment proceedings. He promised Ms. Greene, who was stripped of her committee assignments for making a series of violent and conspiratorial social media posts before she was elected, a plum spot on the Oversight Committee.
He has threatened to investigate the House select committee investigating the Jan. 6, 2021, attack at the Capitol, promising to hold public hearings scrutinizing the security breakdowns that occurred. He has been quietly meeting with ultraconservative lawmakers in an effort to win them over. And on Monday night, he publicly encouraged his members to vote against the lame-duck spending bill to fund the government.
Those entreaties have fallen flat for some of the ultraconservative members of his conference.
In an opinion essay, Representative Andy Biggs of Arizona, who is running as a protest candidate in the speaker race, noted that Mr. McCarthy had said before the midterm elections that he did not see grounds for impeaching any Biden administration officials. Mr. Biggs dismissed Mr. McCarthy’s more recent threat against the homeland security secretary.