In a statement, Ms. McDaniel sought to assuage some of the concerns, stating: “The 168 R.N.C. members choose who will lead the R.N.C. I hope to win their support, and that is the most important endorsement.”
A committee spokesman, Mike Reed, said that Ms. McDaniel and the committee had always followed bylaws not to endorse candidates in Republican primaries. “That policy to remain completely neutral in primaries will continue as long as she is chair,” Mr. Reed said.
Senior Republican officials close to Ms. McDaniel said they were already seeking new arrangements between the R.N.C. and the Trump campaign over the donor and data lists, which would provide Mr. Trump with copies of certain lists but also leave them available to other candidates through the committee. Beyond that, these Republicans said, there are limits to how influential the R.N.C. can be in party primaries.
Ms. McDaniel, a Michigan native, has a gilded political pedigree: She is the niece of Senator Mitt Romney of Utah and the granddaughter of George Romney, a three-term Michigan governor. She earned Mr. Trump’s trust in part by urging him to make trips to her home state during the 2016 campaign, which he credits with helping him win there.
She has told people she does not intend to seek another term after 2022, one person briefed on the discussions said, a move that could ensure her exit before the 2024 presidential cycle gets underway in earnest.
So far nobody has emerged to challenge Ms. McDaniel, but some influential Republicans are trying to stir support for Senator Cory Gardner of Colorado, who has just lost his re-election bid and is well-liked among pro-Trump and Trump-skeptical Republicans alike in Washington. Mr. Gardner did not respond to two emails inquiring whether he had any interest in the chairmanship.
The current and former committee members who may prove most formidable have told associates they will not run. That includes Mr. Barbour — whose uncle, former Gov. Haley Barbour of Mississippi, took over the party after Republicans lost the presidency in 1992 — and Reince Priebus, Mr. Trump’s first chief of staff and a former R.N.C. chairman.