“I tell you what, there’s real trust, that’s why he’s asked me to run again,” Mr. Biden said Monday, dropping this tidbit at the end of a 40-minute conversation, just before he dashed off to his third fund-raiser of the day. “Look, he said, ‘We’re going to run together, are you going to run?’ I said, ‘Of course, you want me to run with you, I’m happy to run with you.’ ”