By Friday evening, concerned about testifying to Congress over the revelations that he discussed wearing a wire to the Oval Office and invoking the constitutional trigger to remove Mr. Trump from office, Mr. Rosenstein had become convinced that he should resign, according to people close to him. He offered during a late-day visit to the White House to quit, according to one person familiar with the encounter, but John F. Kelly, the White House chief of staff, demurred.

Aides began planning over the weekend for his departure, going in to the Justice Department to determine how to recalibrate in the aftermath of it.

Also over the weekend, Mr. Rosenstein again told Mr. Kelly that he was considering resigning. On Sunday, Mr. Rosenstein repeated the assertion in a call with Donald F. McGahn II, the White House counsel. Mr. McGahn — who was dealing with the emergence of another accusation of sexual assault against Judge Brett M. Kavanaugh, the president’s Supreme Court nominee — asked Mr. Rosenstein to postpone their discussion until Monday.

Some White House officials also believed that only the president could legally accept Mr. Rosenstein’s resignation, not Mr. Kelly, according to two people familiar with internal discussions.