“If party leaders see a scenario next winter where Bernie Sanders has a real chance at the Democratic nomination, I think there’s no question that leaders will reach out to Vice President Biden or Secretary of State Kerry or even Gore about entering the primaries,” said Garnet F. Coleman, a Texas state lawmaker and Democratic national committeeman.

Even if none of those Democrats were to announce candidacies this fall, some party officials and strategists suggested that Mr. Biden could be laying the groundwork for an 11th-hour rescue mission during the winter primaries if Mrs. Clinton’s campaign began to implode. Similarly, Mr. Kerry’s friends say they believe he would hear out party leaders if Mr. Sanders appeared likely to capture the nomination and they implored Mr. Kerry, who would have to resign as secretary of state, to try to block him.

The interest in senior statesmen and stateswomen is partly a reflection of the thin Democratic bench after widespread losses in races for governor, Senate and other offices in 2010 and 2014, which has left the party with relatively few experienced, credible presidential contenders, let alone ones willing to take on Mrs. Clinton. (The paucity of fresh faces even gave rise to a joking Twitter hashtag: #Dukakis2016, offering up the party’s 1988 nominee, former Massachusetts Gov. Michael S. Dukakis.)