In response to a CNN article based on four anonymous sources, Ms Warren stood by the reported account of a private meeting between the two senators but said she and Mr Sanders remained “friends and allies”.
“Among the topics that came up was what would happen if Democrats nominated a female candidate,” she said of the December 2018 meeting.
“I thought a woman could win; he disagreed.”
Mr Sanders’ team has strongly denied the claim, with campaign manager Faiz Shakir calling it a “lie”.
“What I did say that night was that Donald Trump is a sexist, a racist and a liar who would weaponize whatever he could,” Mr Sanders said in a statement on the allegation.
“Do I believe a woman can win in 2020? Of course! After all, Hillary Clinton beat Donald Trump by 3 million votes in 2016.”
The dispute has raised concerns among the left-wing of the Democratic Party of a public feud between Mr Sanders and Ms Warren, the only two viable progressive candidates for the 2020 election.
Although both candidates have performed well in polling so far, neither have been able to supplant former vice president Joe Biden as the frontrunner in the Democratic race.
Mr Biden currently averages about 28 per cent in polls, ahead of Mr Sanders (on about 20 per cent) and Ms Warren (on about 16 per cent), according to analysis by RealClearPolitics.
However, Mr Sanders and Ms Warren also share supporters on the Democrats' left-wing, with a Quinnipiac University poll on Monday showing 57 per cent of Sanders supporters would vote for Ms Warren and 52 per cent of Warren supporters would vote for Mr Sanders.
Meanwhile, only 20 per cent and 17 per cent of Biden supporters said they would vote for Mr Sanders and Ms Warren respectively.
In the 2018 meeting, the two candidates are said to have agreed that they would not directly attack each other during the campaign, but that agreement has frayed as the Democratic race has intensified.
Over the weekend, it was reported that the Sanders team has told volunteers to suggest Ms Warren is the candidate for “highly-educated, more affluent people” to convince voters to switch to his campaign, according to a document leaked to Politico.
Ms Warren said she was “disappointed” by the leak, which has not been denied by the Sanders campaign.
Some progressives within the Democratic Party have feared that public disagreements between the two senators could undermine attempts to elect a left-wing candidate for the Democratic nomination.
On Monday, Stephanie Taylor and Adam Green, co-founders of the Progressive Change Campaign Committee (PCCC), said that “a back-and-forth about this private meeting is counter-productive for progressives”.
“In this pivotal moment of the campaign, progressives must work together to defeat Donald Trump and prevent a less-electable establishment candidate like Joe Biden from getting the nomination,” they wrote in a statement.
Waleed Shahid, the communications director for Justice Democrats, told CNN that Mr Sanders and Ms Warren should focus their efforts on defeating candidates such as Mr Biden and avoid encouraging a fight in the progressive movement.
“Warren and Sanders should be focusing their fire on their two corporate-friendly opponents and earning the trust of voters for why their vision will defeat Trump and provide relief for the American people,” Mr Shahid said.