Her role has not been finalized, in part because she is weighing family considerations, but it is expected to be arranged in the coming days, according to Democrats familiar with the discussions.

Mr. Biden has long had a sort of bullpen-by-committee approach to his senior staff, and longtime advisers like Mike Donilon, Steve Ricchetti and his sister, Valerie Biden Owens, are expected to continue playing important roles along with his original campaign manager, Greg Schultz, and deputy manager, Kate Bedingfield.

Addisu Demissie, who managed Senator Cory Booker’s campaign, said the timeline of a presidential contest required any major candidate at this point to start building out the staff and making plans for the general election. A veteran of Hillary Clinton’s 2016 campaign, Mr. Demissie said that it was too late for Mr. Biden to develop a robust operation in big March primary states like Florida, but that there was still time to strengthen his campaign in later-voting states and lay the groundwork for a national race against Mr. Trump.

“You cannot scale up a presidential campaign in any less than five months,” Mr. Demissie said. He suggested that a candidate in Mr. Biden’s position should be “bringing in as much talent as they can and, I assume, not taking the primary for granted but also realizing that the real battle is coming in November and it’s coming pretty damn quick.”

In some regards, Mr. Biden and his top aides have been overwhelmed by his sudden, and unexpectedly broad, success. They are attempting to be sensitive to the staff members who, with little money at their disposal, helped resurrect his candidacy but are also eager to tap into the enormous pool of Democratic talent now available, people close to the campaign said. Hundreds of aides to Senators Amy Klobuchar and Elizabeth Warren, Pete Buttigieg and Mr. Bloomberg are now available for hire.

A day after Mr. Bloomberg’s exit, his top advisers were beginning to map out how to repurpose his sprawling campaign organization to support Mr. Biden and other Democrats in the general election. The former New York City mayor has pledged for months to use his personal wealth to fund efforts against Mr. Trump even if someone else became the Democratic nominee.

Talking to Mr. Biden on the phone Wednesday, Mr. Bloomberg made clear that he would be supportive, but the two did not get into details, according to a Democratic official familiar with the conversation.