But when discussing his family, Mr. Biden sounded somber, and without naming his son Beau, who died on May 30, he spoke of a family in deep grief.

Were he to run, he would “have to be able to commit to all of you that I would be able to give it my whole heart and my whole soul,” he said, “and right now, both are pretty well banged up.”

“I’m not trying to skirt your question,” he said. “That’s the truth of the matter, but believe me, I’ve given this a lot of thought and dealing internally with the family on how we do this.”