In investigating Russian election interference, Mr. Mueller is also examining whether the president tried to obstruct the inquiry. The former F.B.I. director James B. Comey said that Mr. Trump asked him for his loyalty and to end the investigation into his first national security adviser, Michael T. Flynn. After firing Mr. Comey, the president said privately that the dismissal had relieved “great pressure” on him. And Mr. Trump also told White House officials after Attorney General Jeff Sessions recused himself from the investigation that he needed someone running the Justice Department who would protect him.
The experts said the meetings with Mr. McGahn and Mr. Priebus would probably sharpen Mr. Mueller’s focus on the president’s interactions with other witnesses. The special counsel has questioned witnesses recently about their interactions with the president since the investigation began. The experts also said the episodes could serve as evidence for Mr. Mueller in an obstruction case.
“It makes it look like you’re cooking a story, and prosecutors are always looking out for it,” said Julie R. O’Sullivan, a law professor at Georgetown University and expert on white-collar criminal investigations.
She added, “It can get at the issue of consciousness of guilt in an obstruction case because if you didn’t do anything wrong, why are you doing that?”
Central figures in investigations are almost always advised by their own lawyers to keep from speaking with witnesses and prosecutors to prevent accusations of witness tampering. The president has not been questioned by Mr. Mueller; Mr. Trump’s lawyers are negotiating terms of a possible interview. Learning even basic details about what other witnesses told investigators could help the president shape his own answers.
Mr. Trump’s interactions with Mr. McGahn unfolded in the days after the Jan. 25 Times article, which said that Mr. McGahn threatened to quit last June after the president asked him to fire the special counsel. After the article was published, the White House staff secretary, Rob Porter, told Mr. McGahn that the president wanted him to release a statement saying that the story was not true, the people said.
Mr. Porter, who resigned last month amid a domestic abuse scandal, told Mr. McGahn the president had suggested he might “get rid of” Mr. McGahn if he chose not to challenge the article, the people briefed on the conversation said.
Mr. McGahn did not publicly deny the article, and the president later confronted him in the Oval Office in front of the White House chief of staff, John F. Kelly, according to the people.
The president said he had never ordered Mr. McGahn to fire the special counsel. Mr. McGahn replied that the president was wrong and that he had in fact asked Mr. McGahn in June to call the deputy attorney general, Rod J. Rosenstein, to tell him that the special counsel had a series of conflicts that disqualified him for overseeing the investigation and that he had to be dismissed. The president told Mr. McGahn that he did not remember the discussion that way.
Mr. Trump moved on, pointing out that Mr. McGahn had never told him that he was going to resign over the order to fire the special counsel. Mr. McGahn acknowledged that that was true but said that he had told senior White House officials at the time that he was going to quit.
It is not clear how the confrontation was resolved. Mr. McGahn has stayed on as White House counsel, one of the few senior administration officials who has been with the president since the campaign.
Mr. Priebus met with the president in the West Wing in December, according to the people with knowledge of their encounter. Allies of Mr. Priebus, who left the White House in July 2017, have cautioned him to keep his distance. But Mr. Priebus, who is seeking to build a law practice as a Washington power broker who can open doors for clients, has maintained contact and occasionally visited the White House to see Mr. Trump and his own replacement, Mr. Kelly.
Mr. Trump brought up Mr. Priebus’s October interview with the special counsel’s office, the people said, and Mr. Priebus replied that the investigators were courteous and professional. He shared no specifics and did not say what he had told investigators, and the conversation moved on after a few minutes, those briefed on it said. Mr. Kelly was present for that conversation as well, and it was not clear whether he tried to stop the discussion.
It is not illegal for the subject of an investigation to learn what witnesses have told investigators. But that is usually done through lawyers for the people involved because their communications are often shielded from prosecutors because of attorney-client privilege. In organized crime and complex white-collar investigations, prosecutors often ask witnesses whether they have spoken to the person under investigation to determine whether they are coordinating their stories.
Mr. Priebus has had a long and complicated relationship with the president. He was one of the few who publicly defended Mr. Trump after the Times article about his attempt to fire Mr. Mueller, which cited the president’s view that Mr. Mueller had too many conflicts to be the special counsel.
“He expresses concerns with the conflicts, but I never heard the idea or the concept that this person needed to be fired,” Mr. Priebus said last month in an interview on NBC’s “Meet the Press.” “I never felt it was relayed to me that way, either. And I would know the difference between a level 10 situation as reported in that story and what was reality. And it just — to me, it wasn’t reality.”