“Just because something isn’t true doesn’t mean that it can’t cause you harm or damage. I will always protect Mr Trump,” Cohen’s February 13 statement read in part.

Daniels’ attorney, Michael Avenatti, argued that the statement constitutes defamation because Cohen deliberately tried to portray his client as a “liar” and “someone who should not be trusted.”

“Mr Cohen’s statement exposed Ms. Clifford to hatred, contempt, ridicule, and shame, and discouraged others from associating or dealing with her,” Avenatti charged.

The lawsuit emerged after Cohen sent a cease-and-desist letter to Daniels on Monday after the 60 Minutes interview in which she said she was threatened with violence to keep her quiet about her alleged affair with Trump.

Cohen’s lawyer, Brent Blakely, wrote to Daniels’ lawyer to say she had made false and defamatory comments, “namely that he [Cohen] was responsible for an alleged thug who supposedly visited” and threatened her.

“In truth, Mr Cohen had absolutely nothing whatsoever to do with any such person or incident, and does not even believe that any such person exists, or that such incident ever occurred,” Blakely said in the letter.

Daniels, whose real name is Stephanie Clifford, claimed in the   interview on Sunday that after she first attempted to go public with the story, a man approached her and said: “Leave Trump alone.”

Daniels said she was on her way to a fitness class with her infant daughter in Las Vegas when she was accosted in the car park.

“A guy walked up on me and said to me, ‘Leave Trump alone. Forget the story,’” she said.

“And then he leaned around and looked at my daughter and said, ‘That’s a beautiful little girl. It’d be a shame if something happened to her mom.’ And then he was gone.”

In the interview Daniels said she did not know the identity of the man and did not link him to Cohen.

Later, however, when explaining why she signed a document that denied that any affair with Trump had taken place, she said she was told that if she didn’t “they can make your life hell in many different ways”.

Asked who “they” were, Daniels said: “I’m not exactly sure who they were. I believe it to be Michael Cohen.”

And in an appearance on NBC’s Today show on Monday,  her lawyer, Avenatti, said that it had to have been Cohen or someone else involved with Trump.

Cohen’s lawyer, Brent H. Blakely, told Avenatti in a letter Sunday night that statements blaming Cohen for the threat were false and defamatory. 

“In truth, Mr Cohen had absolutely nothing whatsoever to do with any such person or incident, and does not even believe that any such person exists, or that such incident ever occurred,” Blakely wrote. 

Blakely demanded Daniels and Avenatti make clear through the national media “that you have no facts or evidence whatsoever to support your allegations that my client had anything whatsoever to do with this alleged thug.” 

Avenatti responded by taunting Cohen. 

“Will this guy ever come clean with the American people or is he more interested in trying to role play Ray Donovan (badly),” Avenatti told The Los Angeles Times in a text message, referring to the television show about a shady fixer.

On Monday the White House dismissed Daniels’ story, with spokesman Raj Shah telling the daily White House briefing that “The president doesn’t believe any of the claims Ms Daniels made in the interview last night were accurate.” 

When asked if Trump believed Daniels was threatened, Shah said, “No, he does not,” adding that “There’s nothing to corroborate her claim”.

Early on Monday morning, Stephanie Grisham, a spokeswoman for Melania Trump, the president’s wife, tweeted to slam the media for enjoying “speculation & salacious gossip”.

“I’d like to remind people there’s a minor child who’s [sic] name should be kept out of news stories when at all possible,” she wrote.

The number of Americans watching the show more than doubled to 21.3 million, the network said, scoring an 111 per cent increase over last week’s show and the program’s biggest audience since a November 16, 2008 post-election interview with Barack and Michelle Obama. 

Melania was not with Trump in Washington when the show aired. The White House announced that she was would be spending the week in Florida with their son, “as is their tradition for spring break”.

Trump himself stayed silent on the matter, although he tweeted generally about “so much Fake News”.

“Never been more voluminous or more inaccurate,” the president added. “But through it all, our country is doing great!”

Representatives for the president have denied any sexual relationship took place, despite Cohen’s admission that he paid US$130,000 to buy her silence.

Cohen claims to have paid the money out of his own pocket.

Daniels’ lawyer, Michael Avenatti, called that “ludicrous” during the CBS broadcast on Sunday.

The allegation that an individual threatened Daniels with actual violence while attempting to frighten her off going public about her sexual history with Trump takes the dispute between the president and the porn star to an entirely new level.

Up until now the tussle between Trump and Daniels has focused on her claims of a sexual relationship – although Daniels told CBS they had sex only once – and Trump’s denials.

It has also taken in the ongoing legalistic spat over whether a non-disclosure agreement signed by Daniels just 11 days before the 2016 presidential election to silence her was binding or not.

Daniels’ claim of an affair in 2006 bears striking similarities to the televised account given last week by a Playboy model. 

Karen McDougal told CNN that she had had an intense relationship with Trump for almost a year in 2006 and 2007, and that he had once taken her into his home in Trump Tower, Manhattan.

Both McDougal and Daniels allege that their encounters occurred within months of the birth of Barron, Trump’s son with Melania.

A large number of women have also accused Trump of sexual assault. He denies the claims.