She and her attorney Michael Avenatti have teased out details of her relationship with Trump for weeks, with Avenatti giving explosive interviews nearly daily and Daniels taunting the president on social media.
The request, filed in federal court in Los Angeles early Wednesday morning, seeks to require the president to answer questions under oath about the payment, which Daniels' attorney Michael Avenatti describes as "hush money" meant to silence his client and influence the 2016 presidential election, according to CBS News.
Daniels, whose real name is Stephanie Clifford, signed the non-disclosure agreement just days before the 2016 presidential election in return for $130,000.
The president, through spokespeople, has denied the affair, although he has not personally addressed it.
Avenatti's request to question Trump and Cohen is part of Daniels' lawsuit to overturn a contract she signed to keep quiet about what she says was a consensual affair with Trump.
Mr Trump's lawyer's lawyer, David Schwartz, dismissed the motion as "a reckless use of the legal system in order to continue to inflate Michael Avenatti's deflated ego and keep himself relevant".
Trump and Cohen's legal teams are seeking to force Daniels to settle the case in arbitration, which is required under the terms of the agreement. In a Wednesday court filing, he asked the judge in Daniels' lawsuit to set a date for a jury trial, as well as for depositions of Trump and Cohen. Ms Clifford said that in 2011, after Mr Cohen's threat to sue "In Touch", an unidentified man approached her in a vehicle park before a gym class in Las Vegas and said it would be a shame for her daughter if something happened to Ms Clifford. She is now suing Trump to break an agreement meant to prevent her from discussing their alleged affair, claiming that the document is invalid because Trump never signed it.
Avenatti has asked the Trump Organization to preserve all documents connected to Cohen and Clifford since Cohen used his Trump Organization email to make arrangements with Clifford. Avenatti tweeted a photo of a CD-Rom that purported to contain evidence.
But she has since tried to get out the agreement and gave a widely watched interview with "60 Minutes" on Sunday night in which she provided more details of the encounter.
As precedent, the motion references the fact that former President Bill Clinton was deposed while in office in 1998 during the Paula Jones sexual harassment suit.
This combination of file pictures shows U.S. President Donald Trump speaking at the White House on February 14, 2018; and adult film actress Stormy Daniels in Las Vegas, Nevada on February 4, 2018. Daniels also has a pretty hefty $10,000 personal appearance fee for whenever she chooses to feature at strip clubs across the country. She's one of the women who accused Trump of "unwanted sexual touching" and whom Trump had called a liar.