Interviews with nearly two dozen Biden allies and donors over the last week provided the clearest picture yet of the critical early phase in Mr. Biden’s efforts to choose a running mate. It is a process made even more complicated by the fact that Mr. Biden and his team are quarantined in different locations and grappling with how to run a campaign during a national health crisis.

Mr. Biden has stayed in touch with friends, though, and the question of his choice surfaces often on private donor calls. His allies and advisers are also bombarded with outside opinions, and it is something that Mr. Biden, himself a former vice president, cares deeply about.

There is not yet a consensus choice or a sense of who would deliver a surefire boost to the potential ticket, according to those interviewed, many of whom spoke on condition of anonymity, though the two names most frequently mentioned were Senators Kamala Harris and Amy Klobuchar. And his supporters are divided over whether Mr. Biden should prioritize selecting a woman of color as his running mate or whether regional considerations, like ties to the industrial Midwest, should hold greater weight.

“My preference would be a black woman,” said Representative James E. Clyburn of South Carolina, the highest-ranking African-American in Congress and perhaps Mr. Biden’s most crucial supporter. “But I don’t think it has to be a black woman.”

The outbreak of the coronavirus has heightened the stakes of the vice-presidential search. A decision that traditionally reflects a cocktail of political calculations, personal chemistry and governing experience has now taken on outsize importance for Mr. Biden. He would be 78 at inauguration, might not seek a second term should he win, and would be trying to rebuild the country in the wake of a pandemic — factors that make it less likely that he would select someone who offers strong political assets but a relatively thin résumé.