As party leaders gathered for the Democratic National Committee’s summer meeting, a Draft Biden super PAC set up camp in a suite on the 23rd floor of a Hilton hotel. It held briefings on Mr. Biden’s extensive political career and urged activists to keep an open mind about the race. One freebie used to lure people to the suite: chocolate bars with a wrapper that showed a picture of Mr. Biden behind the wheel and a caption that read: “I’m ridin with Biden.”

John Oram, who serves on the DNC’s finance committee, said in an interview that he made a donation to Mrs. Clinton’s campaign and backs her candidacy at this point. If Mr. Biden enters the race, he said, he would likely switch sides.

“He has a good record as vice president, and he’s been clean. No baggage, and he brings something to the party,” Mr. Oram said.