When Mr. Matthews pushed her to say outright that she thought Mr. Bloomberg was lying, she responded, “I believe the woman, which means he’s not telling the truth.”

“Why would he lie?” Mr. Matthews asked.

“Why would she lie?” Ms. Warren responded.

During the debate, Ms. Warren also challenged Mr. Bloomberg over his announcement last week that he was willing to release three women from nondisclosure agreements with his company so they could discuss their complaints about him publicly. As she said on Friday, it was a limited announcement by Mr. Bloomberg, and “not good enough.”

“Let us have the women have an opportunity to speak,” Ms. Warren said. “The Bloomberg corporations and Mayor Bloomberg himself have been accused of discrimination. They are bound by nondisclosures so that they cannot speak. If he says there is nothing to hide here, then sign a blanket release and let those women speak out so that they can tell their stories the way I can tell my story: without having the fear that they’re going to be sued by a billionaire.”

The back-and-forth was a continuation of a confrontation that began during the last debate, when Ms. Warren quoted comments the mayor had reportedly made, describing him as “a billionaire who calls women ‘fat broads’ and ‘horse-faced lesbians.’”

Long before he even became a politician, Mr. Bloomberg had a reputation for crude and sexist comments. In 2012, while mayor, Mr. Bloomberg urged two guests at a party to “look at the ass on her.”

In November, when asked about the comments by The New York Times, a spokesman for Mr. Bloomberg offered a note of contrition, but did not go so far as to apologize.

“Mike has come to see that some of what he has said is disrespectful and wrong,” said the spokesman, Stu Loeser. “He believes his words have not always aligned with his values and the way he has led his life.”