The man, Daniels says, approached her while she was with her baby daughter in Las Vegas, several weeks after she agreed to cooperate with a story that In Touch magazine was preparing about the purported tryst.
That In Touch story never ran, reportedly because a lawyer for Trump threatened to sue the magazine at the time.
Daniels has said the man in Las Vegas who walked up to her said: "Leave Trump alone. Forget the story."
The man then leaned down, looked at Daniels' child and said, "That's a beautiful little girl. It'd be a shame if anything happened to her mom," according to Daniels.
Michael Avenatti, Daniels' lawyer, during an April 17 appearance with her on ABC's "The View," unveiled a drawing of the man that was prepared with a renowned sketch artist.
Avenatti also announced a $100,000 reward for the identification of the man.
A day later, Trump, who had pointedly not tweeted about Daniels or her claims of an affair in the past, referred to the sketch as a "con job" in what apparently was his first public comment about her allegations.
"Mr. Trump's statement is false and defamatory," says Daniels' lawsuit, filed in U.S. District Court in Manhattan under her real name, Stephanie Clifford.
"In making the statement, Mr. Trump uses his national and international audience of millions of people to make a false factual statement to denigrate and attack Ms. Clifford."
Avenatti told CNBC on Monday, "Even if you're the president of the United States, you cannot simply fabricate a story in order to fit your strategic purpose."
"There are serious consequences for that," Avenatti said.
Trump is also being sued in state court in Manhattan for allegedly defaming a former contestant on his TV show "The Apprentice," who claims he sexually groped her.
The president last month lost a bid to have a judge dismiss that case. The plaintiff in that suit, Summer Zervos, says Trump defamed her by calling her and other women who have made claims of sexual misconduct by him liars.
Daniels' new suit, which seeks more than $75,000 in damages, was filed in the same federal courthouse where she appeared two weeks ago for a hearing involving records seized from Trump's longtime personal lawyer, Michael Cohen.
Among those files seized April 9 were ones related to the $130,000 payment he gave Daniels right before the 2016 presidential election in exchange for her signing the nondisclosure agreement about Trump.
Cohen is under federal criminal investigation by prosecutors in New York. He has not been charged.
The Associated Press, in a January story that cited four former In Touch employees, reported that Cohen in 2011 had "sent an email to In Touch's general counsel saying Trump would aggressively pursue legal action if the story [about the alleged affair with Daniels] was printed."
Cohen is a defendant with Trump in Daniels' other pending lawsuit in Los Angeles federal court, which seeks to void the nondisclosure agreement about the alleged affair. Daniels claims the agreement is not valid because Trump never signed it.
In that case, she also claims that Cohen defamed her by allegedly suggesting that Daniels is lying about the purported relationship with Trump.
