“I don’t think this is what Warren wanted to be talking about today,” said Maurice Mitchell, the national director of the Working Families Party, which has endorsed Ms. Warren. “Friends don’t always agree but we believe that Warren and Sanders have far more that unites them — and not just them but their supporters. We think their supporters should focus their fire on the corporate wing of the Democratic Party.”

“A Bernie-Warren battle doesn’t do Bernie or Warren any good,” he went on, “nor our bases, nor our movement. There is a whole corporate wing of the Democratic Party we need to defeat.”

Whoever finishes ahead of the other on the night of the Iowa caucuses will seek to claim the mantle of progressive front-runner, which could have a significant impact on their path to the nomination. Liberals have quietly worried about the acrimony that could surface that evening, if the supporter base of one candidate loudly calls for the other to drop out. Other leaders, who see Mr. Sanders’s and Ms. Warren’s electorates as more distinct, say both candidates will be in contention through the Super Tuesday contests in March.

“This will be a long primary,” Mr. Chamberlain said.

But as the events of the last few days make clear, tensions are growing. And while some dismissed Monday’s war of words as a media-concocted feud, others affiliated with Mr. Sanders accused Ms. Warren’s team of a cynical political move to imperil his reputation.

Early Tuesday morning, Mr. Sanders’s campaign released a new television advertisement focused on his record defending women’s rights. The spot, called “On Our Side,” contrasts Mr. Sanders’s record with Mr. Trump’s. RoseAnn DeMoro, a former leader of the nurses union who is close to Mr. Sanders, said Ms. Warren was making a political play that would backfire.

“I find it unfathomable that she would believe that attacking Bernie Sanders with a lie that he essentially disrespects women — which is just outrageous — would pull the female support toward her,” Ms. DeMoro said. “Women aren’t that shallow.”

“I don’t believe she honestly believes at this stage in the game she can win,” she said.

Senators Kamala Harris of California and Kirsten Gillibrand of New York, both former candidates for president who dropped out of the Democratic primary late last year, weighed in on Tuesday, releasing firm statements refuting the contention at the heart of the dispute.