Daniels, whose real name is Stephanie Clifford, last month sued Michael Cohen, Trump's personal attorney, to be released from the non-disclosure agreement she signed in October 2016 in exchange for $130,000.
The White House has denied that Trump had sex with Daniels. Cohen has said he paid Daniels out of his own pocket.
In Monday's court filing in Los Angeles, Brent Blakely, Cohen's attorney, argued the agreement included a provision that any disputes over it be settled through arbitration, as opposed to open court.
Federal law "dictates that this motion be granted, and that Clifford be compelled to arbitration, as she knowingly and voluntarily agreed to do," Blakely wrote.
Daniels' attorney, Michael Avenatti, said the matter should be settled in open court.
"We will vigorously oppose the just-filed motion by Mr. Trump and Mr. Cohen to have this case decided in a secret arbitration, in a private conference room, purposely hidden from the American public," Avenatti said in a statement.
Avenatti has argued that the non-disclosure agreement is invalid because Trump never signed it. But in Monday's filing Blakely responded that the language of the agreement did not specify that Trump, using the pseudonym David Dennison, needed to sign it for the agreement to be binding.
Blakely also argued that Daniels accepted the $130,000 and did not dispute the agreement for 16 months even though Trump had not signed it.
Daniels has said she and Trump had sex once in 2006 but that they kept in touch for a period of time.
A former Playboy model, Karen McDougal, has described having a 10-month affair with Trump starting the same year, which the White House has said Trump denies. Trump was married to his wife Melania at the time.
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