Mr. Johnson helped circulate a letter, signed by more than three dozen Republican county and state committee leaders, that was addressed to Mr. Trump and called on him to block the endorsement. It states that Mr. Vance is not a registered Republican and provides Mr. Trump with a list of negative comments that Mr. Vance has made against him in the past, including calling him “another opioid” in 2016 and saying that he hoped Mr. Trump would be “soundly defeated” that year.

“While we were working hard in Ohio to support you and Make America Great Again, J.D. Vance was actively working against your candidacy,” the letter says. “He referred to your supporters as ‘racists’ and proudly voted for Evan McMullin in 2016.”

Jeff Mangun, the chairman of the Carroll County Republican Party, who signed the letter and has not endorsed anyone in the Senate primary, echoed the concerns about a potential Trump endorsement of Mr. Vance. “The best-case scenario is for him to remain neutral, but even if he were to endorse someone, I would prefer it would be someone else,” Mr. Mangun said. “I just don’t think it’s a good fit.”

Asked for comment, including about the accusation that Mr. Vance is not a registered Republican, a spokeswoman for the Vance campaign pointed to polling that showed Mr. Vance in second place behind Josh Mandel, a former Ohio state treasurer; a tweet from one Republican Party county chairman denying that he had signed the letter; and another tweet from the anti-abortion group Ohio Right to Life PAC that expressed support for a Trump endorsement of Mr. Vance.

In stump speeches, Mr. Vance has been quick to address the criticism that he has not always been a Trump loyalist, often saying that the best policy is honesty.

“I didn’t like Trump six years ago,” he told a small crowd of supporters this week at a brewery in Hilliard. “I did not think he was going to be a good president. I was very happy to be proven wrong.”

He added, “I was very proud to support the president over the past several years.”

Mr. Trump is said to have been moving toward Mr. Vance for some time, according to the four Republicans familiar with his advisers’ discussions. He has been encouraged by a range of allies and donors, including the billionaire Peter Thiel, who is a primary financial backer of Mr. Vance. The Fox News host Tucker Carlson, who is influential with members of Mr. Trump’s base, has also spoken favorably about Mr. Vance on his prime-time show.