E. Jean Carroll, 75, a television show host and magazine columnist, made the accusation in an essay published on Friday in New York magazine, the local media reported. She said she did not report the alleged attack as a friend of her advised her at the time that she had no chance of winning.
However, the US President said he never met the woman and accused her of making up the allegation "to sell a new book".
More than a dozen women have previously made sexual misconduct allegations against Trump, which he has denied.
In the article, Carroll described meeting Trump in late 1995 or early 1996, in Bergdorf Goodman department store. She said she recognized him as the "real estate tycoon" and that he told her he was buying a present for "a girl".
She said that Trump knew she was a TV agony aunt and the two joked around, encouraging each other to try on some lingerie. Carroll alleged that they then went to a dressing room, where she accused him of raping her.
Trump was around 50 at the time and was married to Marla Maples.
Carroll said she told two friends about the alleged incident, one of whom advised her to go to the police. But according to her, the other advised her against telling anyone saying: "Forget it! He has 200 lawyers. He'll bury you."
The accusation is one of six alleged attacks by "awful men" that Carroll detailed in her article.
Responding to the claim, Trump said in a statement: "I've never met this person in my life. She is trying to sell a new book-that should indicate her motivation. It should be sold in the fiction section."
Trump encouraged anyone with "information that the Democratic Party is working with Carroll or New York magazine" to notify the White House, the BBC reported.
He accused the publication of "peddling fake news".
"Shame on those who make up false stories of assault to try to get publicity for themselves, or sell a book, or carry out a political agenda. It's just as bad for people to believe it, particularly when there is zero evidence," he added.
In his statement, Trump thanked Bergdorf Goodman for "confirming they have no video footage of any such incident".
( With inputs from IANS )