Dick Harpootlian, a former chairman of the South Carolina Democratic Party, had coffee with Mr. Biden at the White House in May, just days before Beau Biden’s death at 46 from brain cancer. They talked through the possibility of launching a campaign, Mr. Harpootlian said. “It’s no secret that Beau wanted him to run,” he said. “If he does what Beau wanted him to do, he’ll run.”

James Smith, a Democratic state representative in South Carolina, said that in a conversation in the last two weeks, Hunter Biden reiterated that he wanted his father to jump in the race. “He feels strongly about his dad running and serving,” Mr. Smith said.

Hunter Biden, who is 45, didn’t respond to requests for comment.

Several Biden allies said they expect a decision to come by August. But some suggested a campaign isn’t likely to be at the top of the vice president’s mind.