“I mean, let’s talk about ‘Medicare for all,’” he said of the single-player health system that Ms. Warren and Mr. Sanders support. “Do you think there’s been any truth in advertising on that? It’s going to raise taxes on middle class people, not just wealthy people.” Mr. Biden proposes a plan that would build on the Affordable Care Act, also known as Obamacare, maintaining a public option for those who need it.

Ms. Warren’s recent advance — combined with the fund-raising difficulties of Mr. Biden’s campaign — have left some of his supporters worried.

Expressing optimism, Mr. Biden said, “I know I’m the front-runner,” adding that he wasn’t concerned about money. “We’re on a course to do extremely well,” Mr. Biden said. “I’m not — I’m not worried about being able to fund this campaign.”

In the last fund-raising quarter, his campaign said it had only $9 million cash in hand, far less than that of his key rivals. But Mr. Biden’s campaign recently reversed course on its longstanding opposition to the assistance of super PACs, opening the door for wealthy supporters to spend unlimited amounts of money to assist his Democratic primary candidacy.

Mr. Biden was pressed by a CBS News reporter on this reversal after a campaign event in Durham, N.C., also Sunday night, and the former vice president pointed to the barrage of attack ads he has faced from Mr. Trump’s allies.

“I learned after the fact from, from my folks, there are a lot of people who said, ‘We can’t let this stand,’” he said “They are able to do that. I haven’t discouraged them from doing it, but I haven’t encouraged it either. I’ve just stayed away. Hands off. But if I’m the nominee and if I win, I promise you I’m going to continue to push for a constitutional amendment to make sure that there’s public funding of elections.”

During his “60 Minutes” interview, Ms. O’Donnell also hammered Mr. Biden about his debate gaffes. While he acknowledged that the Democratic debates had presented a “learning curve” for him, his wife, Dr. Jill Biden, who sat in on part of the interview, expressed confidence that the public would overlook them.