People close to both Mr. Biden and Mr. Sanders say their campaigns see strong political arguments for putting a woman on the ticket, believing such a choice could generate enthusiasm from female voters who make up the backbone of the party and sharpen their contrast with Mr. Trump, who has been accused of sexual misconduct by nearly a dozen women.

Women remain Mr. Trump’s staunchest opponents, with large majorities consistently giving him some of the lowest marks in polling. His presidency has inspired the largest gender gap in history and supercharged turnout among female Democrats.

“The nomination of the vice president is about building unity and a partnership that can best govern,” said Larry Cohen, a longtime adviser to Mr. Sanders and the chairman of Our Revolution, the organization that spun out of the senator’s 2016 presidential campaign. “The women who sought the nomination and others easily meet both criteria.”