But while Mr. O’Rourke has quickly become a beloved figure among many grass-roots activists, some elders in the party argue that Democrats should be careful about swooning too soon.

“We’re always looking for rock stars but I don’t think the country is looking for rock stars,” said Tom Vilsack, the former governor of Iowa and agriculture secretary for President Barack Obama.

Asked specifically about Mr. O’Rourke, Mr. Vilsack suggested it would be odd to nominate a candidate who was just defeated. “It’s probably a good idea to win the first race you were interested in,” he said, adding, tongue-in-cheek, “Maybe he’s the new Abe Lincoln, you lose a race and then run for president and win.”

One of Mr. O’Rourke’s advisers, asked how the Congressman could go from losing a Senate race to running for president, responded by noting that the country had elected a celebrity president in 2016.