The report follows the release of a video from 2005 last Friday in which Trump was caught on a "hot mic" bragging about groping women.
The bombastic billionaire was grilled about those comments during his debate with Democratic opponent Hillary Clinton on Sunday, to which he responded "No... I have not" when asked if he had kissed women without permission or grabbed their genitals.
The Trump campaign denied the New York Times report and called it "character assassination".
Former businesswoman Jessica Leeds, 74, who worked for a paper company, told the Times that Trump groped her on a flight more than three decades ago as they sat next to each other in first class.
About 45 minutes after takeoff, Trump lifted the armrest and began grabbing her breasts and tried to put his hand up her skirt, Leeds said.
"He was like an octopus," she told the newspaper. "His hands were everywhere."
"It was an assault," she said, describing the encounter that occurred when she was 38.
Leeds said a flight attendant had invited her to sit in an empty seat in first class, next to Trump, whom she had never met.
After he groped her, Leeds fled back to her original seat in coach, she said.
Rachel Crooks said she was a 22-year-old receptionist at a real estate company in Trump Tower in 2005 when she encountered Trump outside an elevator in the building one morning.
She said she introduced herself and shook his hand, but he would not let go. Trump began kissing her cheeks and then "kissed me directly on the mouth".
"It was so inappropriate," Crooks told the Times. "I was so upset that he thought I was so insignificant that he could do that."
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Trump turned up at her office days later and asked for her phone number, Crooks said. When she asked why he wanted it, he said he was going to pass it along to his modeling agency.
She said she relented because of the business relationship between Trump and her company, Bayrock Group. The modeling agency never contacted her.
The rest of her time at Bayrock, Crooks said she slipped out of view whenever she saw Trump, and skipped the Trump Organization holiday party to avoid bumping into him.
Neither woman reported the incidents to law enforcement.
The Times spoke with friends and family of the two women, who corroborated their accounts and knew of the incidents before the "hot mic" video was released last week.
The Trump campaign called the article a political attack.
"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," senior communicators advisor Jason Miller said in a statement.
"To reach back decades in an attempt to smear Mr. Trump trivializes sexual assault, and it sets a new low for where the media is willing to go in its efforts to determine this election."
The Times said Trump was highly agitated when questioned about the women's claims, denying them and saying "none of this ever took place."
The paper said he threatened to sue the news organization if the report was published.
