Daniels says Cohen paid her $130,000 12 days before the 2016 election in exchange for her silence about a year-long affair she had with Trump in 2006.
He said their attempts to move the case from the state court in Los Angeles where Daniels filed her claim to a federal court is because that would increase the chances the matter be decided in private arbitration, "thus hiding the truth from the public".
In Friday's filing, the Trump and Cohen attorneys say that since Clifford agreed to pay $1 million for "each breach", she could owe as much as $20 million if an arbitration judge sides with Trump's and Essential Consultants' attorneys.
Attorney Charles Harder - best known for representing Hulk Hogan in his lawsuit against Gawker, which resulted in its bankruptcy - is handling the case on the President's behalf.
Federer eases into quarter-finals at Indian Wells Masters
Meanwhile, No. 31 seed Philipp Kohlschreiber downed Frenchman Pierre-Hugues Herbert 6-4, 7-6 (1) in their fourth-round match. Federer, the defending champion of the competition, won the French player, 7-5 6-4 in just an hour and 22 minutes.
"My client was physically threatened to stay silent about what she knew about Donald Trump", Mr Avenatti told CNN on Friday.
A shot of Mr Trump with Ms Clifford around the time of the alleged affair. She taped an interview with CBS News' 60 Minutes program last week that has yet to be broadcast. "Donald insists he has told her every detail of what allegedly went down between himself and Stormy, but Melania is expecting further details to come to light".
"I think it will become apparent to people when they tune in to "60 minutes" on March 25 as to the details relating to the threat", Avenatti added.
Another shutout expected into gold-medal game with Canada against US
It was a close race between Huckaby and Cecile Hernandez of France with less than half a second separating them. Vernon's Curt Minard finished sixth in the first-ever Para snowboard banked slalom Paralympics event.
Avenatti responded late Friday that the president was engaging in "bullying tactics" aimed at moving the case "behind closed doors, outside of public view and scrutiny".
"The fact that a sitting president is pursuing over $20 million in bogus "damages" against a private citizen, who is only trying to tell the public what really happened, is remarkable".
"We are not going away and we will not be intimidated by these threats".
In a letter to Mr Trump's personal lawyer, Michael Cohen, early last week, Ms Clifford offered to send the $US130,000 to an account nominated by Mr Trump so she could be released from a non-disclosure agreement she signed in October 2016.
India lodges another protest with Pakistan over harassment of its diplomatic staff
New Delhi alleged that its diplomats were facing harassment and being prevented from discharging their duties in Islamabad . Thew two countries have a history of bitter relations and often accuse each other's diplomatic staff of spying.
In bringing Trump directly into the suit, his lawyer Harder appeared to be seeking to answer a key challenge from Clifford's lawyer, Avenatti, that in failing to sign the original deal, Trump was not bound by it. Cohen says he "facilitated" the payment with his own funds, through a company he created, and was not reimbursed by the Trump Organization or campaign. New documents show a to.