Ms. Harris was trying to loosen Mr. Biden’s grip on African-American primary voters, and her searing reference to her own childhood experience with integration might have been the high point of her campaign. But it came at the expense of an older, white candidate who was already fending off questions about his record on matters of race. And Mr. Biden was personally stung by her attack, his advisers said, because he considered her a friend.

Yet Ms. Harris’s surge in the polls did not last, and the two candidates never sparred again in the same way. By October, aides to both Democrats recall, they were getting along well when they ran into each other at the Des Moines airport before heading to Ohio for the debate there.

More significant than their personal rapport, a Harris endorsement of Mr. Biden would be politically useful for both of them.

A 55-year-old woman of African and Indian descent with law enforcement credentials, Ms. Harris was already likely to be on Mr. Biden’s short list, should he emerge as the nominee. Yet she could bolster her chances to be his running mate if she backed his campaign at a critical time, particularly if he did not win in either Iowa or New Hampshire next month and needed a boost in Nevada and South Carolina. And even if she is not chosen for vice president, she would be a leading contender for a cabinet post, such as attorney general.

For Mr. Biden, who is working to consolidate support from Democratic leaders as Senator Bernie Sanders’s progressive candidacy gains strength, an endorsement from Ms. Harris would signal that party leaders were rallying behind his candidacy and offer him a well-known surrogate to stump on his behalf as the race goes on.

The risk for Ms. Harris would be if she were to get behind Mr. Biden only to see him lose in California, which votes on March 3 as part of Super Tuesday. A survey of the state’s Democratic voters, conducted this month by the Public Policy Institute of California survey, found that Mr. Biden was in second place to Mr. Sanders, of Vermont. But the poll highlighted the strength of the progressive bloc in the state: Mr. Sanders and Ms. Warren combined were capturing 50 percent of the vote.

Rose Kapolczynski, a longtime Democratic strategist in California, said Ms. Harris would not damage her prospects for re-election in 2022 by backing Mr. Biden. But if Democrats were to lose the presidency this fall, supporting him could shape how she was perceived by the left, were she to run again for president four years from now.