Mr. Cobb arrived at the White House last summer as some of Mr. Trump’s friends and advisers encouraged the president to fire Mr. Mueller. Mr. Cobb feared that the special counsel was on the verge of issuing subpoenas demanding documents and testimony from White House aides, a move that could have locked the administration in a contentious court fight it was certain to lose. Instead, he repeatedly declared the White House to be in “full cooperation mode.”

He said publicly that the White House had no interest in firing Mr. Mueller, and told friends privately that he would not remain in the administration if Mr. Trump moved to fire the special counsel.

“There is not and will not be any consideration of terminating the special counsel, Bob Mueller,” Mr. Cobb said in an interview in October. “I think the path that he chose of trying to minimize conflict and maximize cooperation is one that benefits the country.”

Mr. Cobb’s hiring appeared to have a soothing effect on the president last year. He repeatedly assured Mr. Trump and the public that the Mueller investigation would end quickly — first by Thanksgiving, then by the start of the new year. Whether that was wishful thinking or an effort to calm an irascible president, Mr. Cobb’s prediction proved incorrect.

Behind the scenes, as Mr. Cobb rushed to turn over records to Mr. Mueller, his relationship soured with the White House counsel, Donald F. McGahn II, who thought Mr. Cobb should have more thoroughly reviewed documents and been willing to assert executive privilege. Mr. Cobb in turn thought that Mr. McGahn’s early response to the investigation had been too slow, bringing the White House to the verge of a subpoena.

Mr. Cobb, a longtime white-collar defense lawyer, was not a supporter of the Trump campaign and never viewed himself as part of that team. He has donated money to both Republicans and Democrats and supported Jeb Bush for president in the 2016 election.

He joked that he had “rocks in his head” as he took the job, and told friends that he expected to stay at the White House only as long as it took to turn over all the information to Mr. Mueller and arrange staff interviews with prosecutors. That job is nearly complete, with only Mr. Trump’s interview with prosecutors still being negotiated.

As for Mr. Flood, he has already had a whiff of the drama that follows Mr. Trump. Following a New York Times report in March that Mr. Trump was in discussions to hire Mr. Flood, the president attacked the article and one of the reporters who wrote it.