The Times article spoke to a number of women who said Trump had touched them inappropriately, which Trump denied. "Your article is reckless, defamatory and constitutes libel per se," Kasowitz wrote.
"We hereby demand that you immediately cease any further publication of this article, remove it from your website and issue a full and immediate retraction and apoplogy."
Trump repeated the threats to sue at his rally in Florida. "Lawyers are doing the due diligence needed to file such a massive suit," a Trump campaign official said on condition of anonymity.
But the Times wasn't intimidated: "We stand by the story, which falls clearly into the realm of public service journalism," newspaper spokeswoman Eileen Murphy said.
McCraw's letter to Kasowitz said that the article would not be retracted.
"The women quoted in our story spoke out on an issue of national importance - indeed, an issue that Mr. Trump himself discussed with the whole nation watching during Sunday night's presidential debate," McCraw wrote. "It would have been a disservice not just to our readers but to democracy itself to silence their voices."
If Trump thought people who criticised him should be silenced, McCraw wrote, the newspaper would be happy to take the matter to court.