Those supporters, in the White House and the Senate, and within the political circles he has moved in for decades, fear that the legacy Mr. Biden has built as an effective partner who took on tough jobs for President Obama, not to mention the deep reservoir of public good will and sympathy he has amassed in his poignant handling of personal tragedies, could be sacrificed in the pursuit of an unsuccessful challenge to Hillary Rodham Clinton for the Democratic nomination.

They fret that Mr. Biden, as well known for his undisciplined, sometimes self-immolating comments as he is for his charm on the trail, could endanger Mr. Obama’s own legacy by injuring Mrs. Clinton’s candidacy and causing his party to lose control of the White House.

“People deeply care about him and admire him,” said one person who is personally close to Mr. Biden and has worked closely with him, speaking on the condition of anonymity because he did not want to jeopardize his relationship with the vice president. “But you obviously have to worry about the feasibility and ultimate impact of a run.”