those four white men are the ones the president goes to for a final gut-check before making a decision.

Mr. Donilon, who polishes Mr. Biden’s speeches and is the “keeper of the flame” when it comes to determining the president’s overall message, is less involved in the day-to-day West Wing operations than David Axelrod, who performed a similar role for Mr. Obama. But he remains an influential force, often prodding Mr. Biden toward a conclusion. He tends to stay mostly silent until the very end of a discussion, at which point Mr. Biden often embraces whatever point he has made.

“I agree with Mike” signals the end of the meeting, according to people who have witnessed exchanges between the two men.

Mr. Klain has the most regular contact with the president, with a standing daily Oval Office meeting and a mandate to keep Mr. Biden’s agenda moving forward. He has been a constant in the president’s meetings with his coronavirus team as he maps out the administration’s operational response. He is also the lone Twitter obsessive in Mr. Biden’s inner circle, amplifying reporters when he agrees with them, and questioning them when he does not.

Mr. Reed weighs in sporadically with treatises on the issues he believes voters most care about — his ideas, aides say, shape the arc of Mr. Biden’s most important speeches.

Mr. Ricchetti, who led Mr. Biden’s 2020 presidential campaign and has deep connections to Capitol Hill, is the designated optimist in the group. He is the president’s golfing buddy and the person most often described as a genuine “F.O.P.,” or friend of the president. Last month in Wilmington, Del., Mr. Biden’s first round of golf as president was with Mr. Ricchetti and the father-in-law to Mr. Biden’s son, Beau Biden.

Mr. Ricchetti is also in charge of helping the president sort out another consequential decision: which of his allies will receive ambassadorships that are crucial to preserving the interests of the United States. Initially, the White House said that Mr. Biden would be making his first round of decisions in mid-April.

The president is already well past that deadline. On May 4, Ms. Psaki told reporters that the president would be evaluating nominees “soon.” Asked to define “soon” — Days? Months? Weeks? — Ms. Psaki said out loud what many of the president’s aides were no doubt thinking.

“Well,” she said, “I think it depends on when the president makes some decisions.”

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