The lawyers filed two motions on Friday at the US District Court in Los Angeles in a public legal fight that Stephanie Clifford, whose stage name is Stormy Daniels, started last week, reports The New York Times.
Clifford has sued Trump to get out of an agreement that she had struck to be paid $130,000 to stay silent about the affair that started in 2006.
Trump also formally joined his legal team's response to Clifford's suit in a motion to move the case from state court in Los Angeles she had filed her claim, to federal court.
Until now, Trump had kept his distance from the case, leaving it to his personal lawyer, Michael Cohen, to take the lead in refuting Clifford's claims.
When Cohen brought a temporary restraining order seeking to silence Clifford in late February, he did so on behalf of a shell company, Essential Consultants, he used to pay her through, not on behalf of Trump, reports The New York Times.
Friday's filing comes a day after news broke that CBS News' show "60 Minutes" was planning to run a segment featuring Clifford and her lawyer, Michael Avenatti, on March 25.
Avenatti posted a series of tweets late Friday about the potential damages, calling the new filing "another bullying tactic from the President and Cohen", CNN reported.
"They are now attempting to remove this case to federal ct in order to increase their chances that the matter will be decided in private arbitration, thus hiding the truth from the public.
"The fact that a sitting president is pursuing over $20M in bogus 'damages' against a private citizen, who is only trying to tell the public what really happened, is remarkable. Likely unprecedented in our history. We are not going away and we will not be intimidated," he tweeted.
Clifford and Trump allegedly had an affair in 2006, which is the focus of the nondisclosure agreement, according to her lawsuit against Trump.
Cohen has said Trump "vehemently denies" any affair took place.
Clifford's attorney said Cohen has continued to intimidate his client into keeping silent.