It was past midnight on June 28, and four podcast hosts were wide awake in a hotel suite in Boston.

Hours earlier, Democrats around the nation had gone to bed stunned after President Biden fumbled through his debate against former President Donald J. Trump. Crowded around a table in a dimly lit room, the four men, hosts of the popular podcast “Pod Save America,” were trying to process what they had just seen — not only for themselves but also for their millions of listeners.

“It would be silly not to have this conversation,” one of the show’s hosts, Jon Favreau, said on the recording.

“A Brutally Honest Debate Recap,” the 892nd episode of the seven-year-old political podcast hosted by four former Obama administration officials, was a turning point in what Democrats were willing to say about Mr. Biden’s chances in the 2024 race.

For months, the hosts had acknowledged polling and reporting that showed Mr. Biden’s age was a sticking point for voters. Last year on the show, they hosted Dean Phillips, the Minnesota congressman who had wanted to challenge Mr. Biden for the Democratic nomination largely because of concerns about his age.

But like most mainstream, high-profile Democrats, the men of “Pod Save America” had stopped well short of suggesting Mr. Biden should step aside.

Before the post-debate episode aired, anyone discussing the possibility of replacing Mr. Biden was seen as a “bad Democrat,” said Alyssa Cass, a Democratic strategist who has been a guest on one of the hosts’ spinoff podcast, “Pollercoaster.” But coming from “Pod Save America,” the case for a new candidate wasn’t so easily dismissed.