Though Mr. Biden said that Mr. Sanders had been “very honest” about the costs of Medicare for All, he still drew a forceful rebuke from the Vermont senator, who accused Mr. Biden of echoing the insurance and pharmaceutical industries — and Republicans — in his criticisms of Medicare for All.

And on Monday, Mr. Sanders blasted Mr. Biden’s health care plan, suggesting it was aligned with “corporate greed.”

After initially failing to respond to requests for comment, a Biden spokesman, Andrew Bates, sent a statement after this article was published saying that Mr. Biden “disagrees with any Republican, Democrat, or Independent who wants to dismantle the ACA at a time when we should be protecting and building on it.”

Mr. Sanders is running in the Democratic primary but is an Independent from Vermont.

Aides to Mr. Sanders suggest the escalating confrontation plays into his hands: They have been trying to frame the primary as a competition between him and Mr. Biden, hoping to set up the same kind of insurgent-versus-establishment contrast that worked to Mr. Sanders’s advantage in 2016, when he was the foil to Hillary Clinton. And they view poaching supporters from Mr. Biden, particularly working-class Democrats, as the most promising way for Mr. Sanders to grow his base.

To help their case, they have been urging Mr. Sanders to go after Mr. Biden more aggressively.

Yet for weeks, they have remained frustrated with polls showing Mr. Biden still leading Mr. Sanders by comfortable margins, some allies said. The recent surges by Ms. Warren and Ms. Harris and the fund-raising prowess of Mayor Pete Buttigieg have also dispelled the notion that it is a two-person contest.

“He’s clearly losing support, both in the national polls and in states,” Stan Greenberg, a veteran Democratic pollster, said of Mr. Sanders. “He has no choice. I think he’s made a decision, he’s got to make himself relevant again. Health care was the issue in which he was relevant last time, and it was very important in the campaign in 2016, so I think it makes a lot of sense for him to try to draw this contrast on health care, particularly in the primary.”