A lawyer for Michael Cohen's Essential Consultants limited liability company, which paid Daniels $130,000 under the agreement, made the filing, which said that Daniels violated the deal "at least" 20 times.
"The Settlement Agreement provides for liquidated damages of one million dollars [US] per instance of breach," the court filing says.
Daniels signed the agreement pertaining to an alleged affair a decade earlier shortly before the 2016 election, leading to allegations that the payment amounted to an illicit contribution to Trump's campaign.
Lawyer Michael Avenatti filed a lawsuit on behalf of Daniels, whose real name Stephanie Clifford, earlier this month seeking to toss out the confidential settlement.
2008 file photo, adult film star Stormy Daniels Source: AAP
The suit alleged that Daniels began an "intimate relationship" with Trump in the summer of 2006 that continued well into 2007.
On July 12, court proceedings will get under way to decide whether Daniels can dissolve the agreement.
The White House has denied any sexual encounter between Trump and Daniels, who has offered to return the $130,000 she received so that she can be free to "speak openly and freely about her prior relationship with the president and the attempts to silence her."
She is also asking to be allowed to publish text messages, photos and videos relating to the president, Avenatti said in a letter to Cohen.
'Bullying tactic'
Using the pseudonyms Peggy Peterson and David Dennison, Daniels and Trump were to sign the agreement, along with Essential Consultants.
But Trump never signed, providing the basis for Avenatti's attempt to have it thrown out and release Daniels from her obligations.
Avenatti took to Twitter on Friday to criticise the court filing.
He added the filing is a "bullying tactic" from Donald Trump and his lawyer.
According to the court filing, Essential Consultants began an "arbitration proceeding" as provided for by the agreement at the end of February.
Trump has ample reason to want to keep the affair quiet.
The issue could hurt the president with religious voters, who were key to his electoral success, and appears to have already caused strains between the president and First Lady Melania Trump.
Physical threat to Daniels?
Earlier Friday the White House condemned any threat against Daniels after the porn star's lawyer claimed she was threatened with physical violence if she revealed details of the alleged tryst.
"Obviously we take the safety and security of any person seriously," said press secretary Sarah Sanders.
The White House "certainly would condemn anyone threatening any individual, but I have no knowledge of that situation," she said.
White House officials have tried to deflect questions about the alleged affair, pointing to previous denials by the Trump campaign.
