The woman, lawyer Elizabeth Beck, told CNN's New Day she asked to take a break while questioning Trump, 69, in a lawsuit over a failed Florida real estate project. When Trump objected, she explained it was "urgent" and took out a breast pump to show it to the group of lawyers in the room, The New York Times reports.
The billionaire reality TV star currently leads the GOP presidential field, with 20 percent support.
"He got up, his face got red, he shook his finger at me and he screamed, 'You're disgusting, you're disgusting,' and he ran out of there," Beck said in the CNN interview.
Trump, who saw the CNN interview with Beck, remembers the incident differently.
"I watched that and I thought that was disgraceful," Trump told CNN's Dana Bash. "She's a terrible attorney. We beat her soundly. She's got a terrible reputation. Bottom line, I beat her – and – it was a breast pump. She wanted to breast pump in front of me. I may have said it was disgusting. I thought it was terrible."
Trump's attorney, Allen Garten, who was present at the time, doesn't dispute that Trump called Beck "disgusting" but said Beck was indeed attempting to pump in plain sight.
"She was attempting to breast feed – to pump in the middle of a deposition, in a deposition room with five lawyers and was not excusing herself," he told CNN by phone.
Despite, or perhaps because of, the controversy, Trump's popularity continues to rise. A Quinnipiac University national poll, released Thursday morning, indicates Trump leads the GOP presidential field by a significant margin.