Democratic presidential candidate Elizabeth Warren claims rival Bernie Sanders told her a woman could not win this year's presidential election.
Ms Warren said the comment was made during a two-hour meeting with Mr Sanders in 2018.
Mr Sanders denied the claim and insisted he believed a woman could beat President Donald Trump.
Mr Sanders and Ms Warren are the most prominent progressive candidates for the Democratic nomination.
Their spat has left an unstated non-aggression pact between them in tatters and is likely to come up at a live televised debate on Tuesday night in Des Moines, Iowa.
How has Sanders responded?
CNN quoted Warren aides as saying she had been told by Mr Sanders during a private meeting that a woman could not win.
In his response, he said the comment had been made up by "staff who weren't in the room and are lying about what happened".
The Vermont senator told CNN: "It is ludicrous to believe that at the same meeting where Elizabeth Warren told me she was going to run for president, I would tell her that a woman couldn't win.
"What I did say that night was that Donald Trump is a sexist, a racist and a liar who would weaponise whatever he could.
"Do I believe a woman can win in 2020? Of course! After all Hillary Clinton beat Donald Trump by 3 million votes in 2016."
But later on Monday, Ms Warren stood by the allegation. "Among the topics that came up was what would happen if Democrats nominated a female candidate," she said in a statement.
"I thought a woman could win, he disagreed." The Massachusetts senator added: "I have no interest in discussing this private meeting any further because Bernie and I have far more in common than our differences on punditry."
She added that the two remained "friends and allies".
It took a while, but it appears that the much-anticipated showdown between the two most liberal candidates in the Democratic presidential field is happening at last.
For the entirety of 2019, as both candidates campaigned and debated on similar issues, the unspoken "non-aggression pact" had held, largely because both candidates are well liked by their supporters and any attack could backfire. Less than three weeks from the Iowa caucuses, however, and with Ms Warren and Mr Sanders neck-and-neck in the polls, the tension is approaching a breaking point.
Ms Warren's assertion that Mr Sanders told her he didn't believe a woman can win could prove damaging to him among those who backed Hillary Clinton for president in 2016 - and view Mr Sanders as having damaged her campaign during his primary race against her. Ms Warren's Sunday warning that Democrats should not return to the "factionalism of 2016" seems designed to rekindle these resentments.
All eyes will be on Ms Warren and Mr Sanders as they take to the debate stage here in Des Moines on Tuesday night. Even if they largely hold their fire, however, political trench warfare between Sanders and Warren supporters in Iowa and elsewhere as Democrats prepare to caucus and vote seems unavoidable now.
Is this the first Warren-Sanders clash?
No. On Sunday, Politico reported that some Sanders volunteers had been instructed to depict Ms Warren as an elitist.
The volunteers were reportedly telling voters on the doorstep that Ms Warren's appeal was limited to "highly educated, more affluent people".
Ms Warren accused Mr Sanders of "sending his volunteers out to trash me". She warned against the kind of internecine "factionalism" she said split Democratic voters in 2016.
He denied sanctioning the script.
Mr Sanders told reporters in Iowa: "We have hundreds of employees. Elizabeth Warren has hundreds of employees.
"And people sometimes say things that they shouldn't. You have heard me give many speeches. Have I ever said one negative word about Elizabeth Warren?"
The pair will be among six candidates on stage in Tuesday's televised debate hosted by CNN, including former Vice-President Joe Biden and South Bend Mayor Pete Buttigieg.
According to an opinion poll taken on 8-9 January, 23% of registered Democrats said they supported Mr Biden, while 20% supported Mr Sanders and 15% Ms Warren.
Former Democratic presidential candidate Kirsten Gillibrand, who last August ended her campaign focusing on women's equality, has commented on the gender row.