"Trump used his national and worldwide audience of millions of people to make a false factual statement to denigrate and attack Ms Clifford", the filing to a NY federal court reads.
In the tweet earlier this month, the president said: "A sketch years later about a nonexistent man".
Additional reporting from Newsy affiliate CNN.
Cohen faces a criminal probe into his business affairs by federal prosecutors in Manhattan, including over the $130,000 payment, which he has called legal.
Mr Trump's lawyers did not immediately respond to requests for comment. "It'd be a shame if something happened to her mom", Daniels claims. The White House has denied her allegations.
Daniels alleged that a man approached her outside a fitness class she was attending with her infant daughter and delivered the threat, which shook her.
Daniels, whose real name is Stephanie Clifford, lodged the complaint with the US District Court for the Southern District of NY.
Prosecutors seize on President Trump's comments about Cohen on 'Fox & Friends'
I just want you to remember that this is what Trump volunteered to his friendly media mouthpieces. Trump knew nothing about the agreement with my client, the $130k payment, etc".
Trump has already faced down another lawsuit related to his Twitter use, one that says his blocking of others on Twitter violates the blocked users' First Amendment rights because it prevented them from communicating with the President.
Trump presently has 51.4 million Twitter followers.
Daniels sued Trump as an individual, not in his capacity as president, which could pose problems for any defense.
"Mr. Trump's statement is false and defamatory", according to the lawsuit.
In a separate story in the same issue, the Enquirer also claims that Trump has taken and passed a polygraph test "proving no Russian Federation collusion!"
Trump has argued that he was immune from that lawsuit.
Libowitz speculated the Trump camp is footing Cohen's legal bills because the President is anxious about what his longtime fixer could potentially tell investigators if they offer him a plea deal.
The case is Clifford v Trump, U.S. District Court, Southern District of New York, No. 18-03842.