Then, in the weeks after it became clear that Mr. Biden was the party’s near-certain nominee, Mr. Obama — telling a friend he needed to “accelerate the endgame” — had at least four long conversations with his former vice president’s remaining rival, Senator Bernie Sanders. Mr. Obama’s efforts to ease the senator out of the race played a significant role in his decision to end his bid and, on Monday, endorse Mr. Biden, according to people close to the Vermont independent.

By that time, Mr. Biden and Mr. Obama had already begun hashing out the thorny questions of how, when and where to deploy a former president thrust into an unfamiliar role as his sidekick’s sidekick. Mr. Obama endorsed Mr. Biden in a video released on Tuesday