In the hourlong conversation with Richard A. Harpootlian, the former chairman of the South Carolina Democratic Party, the vice president did not reveal which way he was leaning, according to a Democrat familiar with the conversation who spoke on the condition of anonymity to discuss a private call. Mr. Biden described the recent death of his son Beau as an “open wound” that had not begun to heal, and was grappling with whether he could devote his full energy to a campaign.

But at the same time, Mr. Biden and those who support him are moving to put the pieces into place for a possible candidacy. The vice president directed Mr. Harpootlian to get in touch with one of his closest political advisers, Mike Donilon, and has been calling other supporters. And he has permitted his advisers to discreetly contact operatives in early nominating states to determine how fast they could organize a campaign.

A once bare-bones “Draft Biden” movement has entered a new, more aggressive phase that resembles an exploratory committee. With a Biden family loyalist now involved, the group is contacting Democratic insiders about everything from how to establish a presence at the Iowa State Fair this week to which Democratic donors and officials in South Carolina need to be contacted.