The release that Ms. Flores drafted did not go out.

But the fact that Mr. Rosenstein may be on the job for at least another 72 hours is unlikely to be the end of the story. A departure by Mr. Rosenstein this week would thrust the administration into further turmoil only weeks before the midterms.

As the top Justice Department official overseeing the investigation by Robert S. Mueller III, the special counsel, Mr. Rosenstein has long been the target of Mr. Trump’s bitter grievance about what he calls a politically motivated witch hunt. Mr. Rosenstein has repeatedly backed Mr. Mueller.

Though officials have said their relationship has improved recently, the president was said to have considered terminating Mr. Rosenstein in the summer of 2017. More recently, in a Twitter rant in April, Mr. Trump accused Mr. Rosenstein of being one of the most conflicted officials at the Justice Department, asserting without evidence that he was among those seeking proof of a Trump-led conspiracy with Russia’s election interference.

“No Collusion, so they go crazy!” Mr. Trump wrote.

If Mr. Rosenstein leaves, Noel J. Francisco, the solicitor general, would assume oversight of the Russia investigation, according to a Justice Department official. Matthew G. Whitaker, Mr. Sessions’s chief of staff, would become acting deputy attorney general, an unusual move; typically, a top aide to the deputy attorney general would take over the post.

Critics have said that Mr. Francisco cannot oversee the Russia investigation without a waiver from the White House because his former law firm, Jones Day, is representing the Trump campaign in the inquiry, creating a conflict of interest. Justice Department officials have not addressed whether a waiver would be needed if Mr. Rosenstein departs.

Republican lawmakers aligned with Mr. Trump have spent months wrangling over information pertaining to Justice Department investigations. Democratic opponents have said that those increasing demands were meant to corner Mr. Rosenstein and eventually push him to either compromise the integrity of the investigations or to resign.

Representative Robert W. Goodlatte, Republican of Virginia and the chairman of the House Judiciary Committee, appeared to validate Mr. Rosenstein’s concerns that lawmakers were planning to scrutinize the events described in the Times article. It was based on interviews over several months with people who were told about Mr. Rosenstein’s comments at the time or who were briefed on memos that documented them, including some written by Andrew G. McCabe, then the acting director of the F.B.I.

Mr. Goodlatte said that he planned to issue a subpoena for Mr. McCabe’s memos as soon as this week. House Republicans close to Mr. Trump had already made one attempt to obtain copies of them but were rebuffed by the Justice Department.