“He’s a friend,” Mr. Biden continued. “We’re competitors. He’s a friend. I don’t want him to think I’m being presumptuous, but you have to start now deciding who you’re going to have background checks done on as potential vice-presidential candidates, and it takes time.”

Mr. Biden has said previously that his staff has been in touch with Mr. Sanders’s team, and in an interview with MSNBC’s Brian Williams on Tuesday said that “there ought to be a way we could accommodate his concerns on other matters in terms of everything from people being engaged, to his organization.”

In a statement earlier this week, the Biden campaign said that “both teams remain in touch as the candidates address the coronavirus crisis and its impacts on our economy.”

Mr. Sanders, for his part, has said that he sees a “narrow path” in the race, but that campaigns are also an “important way to maintain that fight and raise public consciousness” about key issues. The Vermont senator is popular among younger and more progressive voters who are skeptical of Mr. Biden, and Mr. Biden has sought to avoid appearing to pressure Mr. Sanders to exit the race, emphasizing that it is his decision.

But at the fund-raiser Friday, Mr. Biden made clear that he was looking ahead to a potential future administration, speaking in some of his clearest terms so far about what that team could look like should he win. He said that by “sometime in the middle of the month we’re going to announce a committee that’s going to be overseeing the vice-presidential selection process,” and said — as he has before — that he has discussed the search process with former President Barack Obama.