Mr. Trump’s lawyers are expected to try to set ground rules for any interview or provide answers to written questions. If Mr. Trump were to refuse outright to cooperate, Mr. Mueller could respond with a grand jury subpoena.

Image Ty Cobb, the senior White House lawyer on the case, pledged full cooperation with Mr. Mueller’s investigation. Credit... Gabriella Demczuk for The New York Times

The White House had no comment on the discussions about a possible interview, which were first reported by NBC News.

One person familiar with the discussions said Mr. Mueller appeared most interested in asking questions about the former national security adviser, Michael T. Flynn, and the firing of the F.B.I. director, James B. Comey — not the broader question of possible collusion with Russia. Those topics signal an interest in whether Mr. Trump tried to obstruct justice. The person was not authorized to talk about internal discussions and spoke on condition of anonymity.

The obstruction investigation focuses on whether Mr. Trump broke the law by asking Mr. Comey to end the investigation into Mr. Flynn and whether he fired Mr. Comey to try to hinder the F.B.I. investigation into Russia-related matters. Shortly after dismissing Mr. Comey in May, the president told Russian diplomats in an Oval Office meeting that doing so had relieved “great pressure” on him.

Mr. Trump has sat for depositions before and shown discipline when under oath. His testimony in civil cases reveals a canny ability to avoid being cornered and a frank acknowledgment that he uses “truthful hyperbole” or “innocent exaggeration.” But he has never faced questioning from someone like Mr. Mueller, a veteran prosecutor and former F.B.I. director who has a dozen experienced litigators behind him.