Aides say they cannot really imagine a situation in which Mr. Biden would run against Mrs. Clinton. But Mr. Biden, 70, is keenly sensitive to perceptions about his political stature, and he is sending signals that he wants to be taken seriously in the 2016 conversation. He also wants to maintain maximum leverage within the Democratic establishment, especially with the Clintons, three people close to him said.

“Joe always says, ‘If you’re not on your way up, you’re on your way down,’ ” one of the people said. A former administration official aide agreed, saying, “He needs to make people think he has skin in the game in the future so that they treat him relevant now.”

During Mr. Obama’s first term, Mrs. Clinton and Mr. Biden often found themselves agreeing on foreign policy issues. They did clash on the surge in Afghanistan in 2009, which Mr. Biden opposed and Mrs. Clinton and Secretary of Defense Robert M. Gates supported. But they often found more common ground than either of them had with Mr. Obama, and they ended up trying to bring Mr. Obama closer to their positions.

During the 2011 protests in Egypt, Mr. Biden and Mrs. Clinton both took a “devil you know” approach and, more than Mr. Obama did, saw benefits to keeping President Hosni Mubarak in power.