The dispute arose when 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.”
He 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.”
“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 in a statement. “Among the topics that came up was what would happen if Democrats nominated a female candidate. I thought a woman could win, he disagreed,” she said.
She added that the two remained “friends and allies”.
BBC North America reporter Anthony Zurcher said the rising tension has left some progressives concerned that both candidates could be weakened in the end.
The pair are preparing to take part in a televised Democratic debate in Iowa on Tuesday. Six candidates have qualified to take part 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% said they would vote for Ms Warren.
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Media captionJoe Biden embarks on his ‘No Malarkey’ bus tour