In a series of tweets this month, Mr. Trump criticized Mr. McConnell publicly, then berated him in a phone call that quickly devolved into a profane shouting match.

During the call, which Mr. Trump initiated on Aug. 9 from his New Jersey golf club, the president accused Mr. McConnell of bungling the health care issue. He was even more animated about what he intimated was the Senate leader’s refusal to protect him from investigations of Russian interference in the 2016 election, according to Republicans briefed on the conversation.

Mr. McConnell has fumed over Mr. Trump’s regular threats against fellow Republicans and criticism of Senate rules, and questioned Mr. Trump’s understanding of the presidency in a public speech. Mr. McConnell has made sharper comments in private, describing Mr. Trump as entirely unwilling to learn the basics of governing.