“They began as friendly rivals,” said David Axelrod, a longtime adviser to Mr. Obama, recalling the primary. “But the relationship has been forged in the fires of many tests. There’s a real bond between them.”

The two men often spend several hours a day together when both are in town, in addition to a weekly one-on-one lunch, and officials say Mr. Biden is almost always the last person in the room with the president.

Mr. Biden’s decision not to ask for a dedicated portfolio of issues, as Vice President Al Gore did under President Bill Clinton, prompted skeptics to predict he would lack influence. But Mr. Biden has become the president’s chief troubleshooter, shepherding a stalled arms-reduction treaty with Russia through the Senate, for example. He has also been his point person on issues ranging from Iraq to budget negotiations with Congress, which collapsed this week over disagreements about taxes and spending cuts.