Ms Crooks, who at the time was a 22-year-old receptionist at Bayrock Group which is based in Trump Towers, said that she introduced herself to the billionaire businessman.
She claimed they shook hands but Mr Trump would not let her go and began kissing her on the cheek. She said he then "kissed me directly on the mouth".
"It was so inappropriate. I was so upset that he thought I was so insignificant that he could do that," she told the New York Times.
In a statement, the Trump campaign said the claims were "fiction", describing the allegations as a "character assassination".
"This entire article is fiction, and for the New York Times to launch a completely false, coordinated character assassination against Mr Trump on a topic like this is dangerous. To reach back decades in an attempt to smear Mr Trump trivialises sexual assault, and it sets a new low for where the media is willing to go in its efforts to determine this election," said Jason Miller, Mr Trump's senior communications advisor.
"It is absurd to think that one of the most recognisable business leaders on the planet with a strong record of empowering women in his companies would do the things alleged in this story, and for this to only become public decades later in the final month of a campaign for president should say it all.
Mr Trump told the newspaper: "None of this ever took place."
Another newspaper, The Palm Beach Post in Florida, reported on Wednesday night that a woman said Mr Trump groped her at his Mar-a-Lago estate 13 years ago. Mr Trump's campaign said her allegation "lacks any merit or veracity."
Mindy McGillivray, 36, said she did not report to authorities her 2003 encounter with Mr Trump at the time but had shared the story with close friends and family. A man who had accompanied her to Mar-a-Lago that day, Ken Davidoff, told the newspaper he vividly remembers McGillivray telling him Mr Trump had groped her.
Mr Davidoff said he had brought Ms McGillivray with him when he joined his father to shoot pictures during a concert by singer Ray Charles on Jan 24, 2003. After the show, as people were saying goodbye to Charles, Ms McGillivray felt "a pretty good nudge, more of a grab" close to the centre of her bottom, she told the newspaper.
"I turn around and there's Donald. He sort of looked away quickly. I quickly turned back, facing Ray Charles, and I'm stunned," she said.
Ms McGillivray said she considered making a scene at the time but decided "to stay quiet". Mr Trump's remarks at the second presidential debate in which he denied groping women changed her mind, she said.
During the debate on Sunday night (local time), Mr Trump sought to draw a distinction between his "locker room banter" and what he claimed were the "actions" of Bill Clinton, the former president and husband of his White House rival Hillary Clinton.
"Mine were words - his were actions," he said.
"So, don't tell me about words. I am absolutely, I apologise for those words. But it is things that people say.
"Bill Clinton. He was impeached. He was no longer allowed to practice law. He had to pay an $850,000 fine to one of the women, Paula Jones, who is here tonight.
"That when Hillary brings up a point like that, brings up words I say 11 years ago, I think it's disgraceful. And I think she should be ashamed of herself, if you want to know the truth."
