People familiar with Mr. Trump’s thinking have long described private conversations with the president in which he has said he is eager to meet with Mr. Mueller, a product of his belief that he can sell or coax almost anyone into seeing things his way.

“I would love to do that — I’d like to do it as soon as possible,” the president told reporters on Wednesday of the prospect of being interviewed by Mr. Mueller, adding that his lawyers have told him it would be “about two to three weeks” until it takes place.