Choosing Ms. Harris, 55, would not only provide a gender balance but also would add a black woman to the ticket after black voters helped revive Mr. Biden’s campaign in February. But as Jill Biden’s recent criticism indicated, the memory of Ms. Harris’s debate stage attack may hinder her chances.

“I have to tell you that I’m a little torn in terms of my choices,” said Alma Gonzalez, a D.N.C. member from Florida. “If it were me and if I was Joe Biden, I would say to Senator Harris, ‘Do you want to be on the Supreme Court or be my vice president?’”

Presidential candidates rarely place public restrictions on their pick, preferring to keep options open so they can pivot their selection to suit the shifting dynamics of the campaign. Veterans of past vice-presidential searches said the most important elements have been how comfortable the nominees are with their would-be partners.

And while past campaigns spent months vetting candidates and agonizing over running-mate strategy, there’s very little academic research suggesting that the vice-presidential pick has a huge impact on winning the general election.

“The first and most important criteria is, can this person help you win in November and will they at least not hurt you in November,” said Mr. Podesta.

For Mrs. Clinton, that meant ruling out candidates from states with Republican governors, like Senator Sherrod Brown of Ohio. If she won, her team feared that Mr. Brown could be replaced in the Senate by a Republican and shift the balance of the chamber away from her future administration.

Unlike any nominee since Mr. Gore, Mr. Biden has a unique view into the selection process, having gone through it himself. While Mr. Obama started with a list of 20 candidates, he faced pressure to select Mrs. Clinton as his running mate and create a “unity ticket.” After Mr. Obama rejected that idea, the choice came down to a “coin toss” between Mr. Biden and Senator Evan Bayh of Indiana. Mr. Biden was more energetic and enthusiastic in his interview, according to aides.