The letter also asked that the president's lawyers agree that neither Trump nor the shell company Cohen used to pay Clifford would attempt to block the broadcast of an interview Clifford taped with CBS News'"60 Minutes" program last week.
Cohen, did not immediately respond to a request for comment on the letter, nor did Cohen's attorney.
Clifford proposed on Monday that she be allowed to discuss her alleged affair with the president freely after repaying the money. While Trump's camp has denied the relationship, Daniels's lawyers have said the affair did happen and that the non-disclosure agreement was invalid because it was not signed by Trump. Cohen did not explain why he made the payment or say whether Trump was aware of it.
Clifford in turn sued Trump for the right to speak about the "intimate relationship" she claims they had, saying he never signed the agreement to keep her quiet. It was the first time the White House had admitted the President was involved in any way with Clifford.
Clifford could face a US$1 million penalty if she breaks the nondisclosure agreement. It's unclear when or if that interview will air, but lawyers for the president are considering mounting a challenge to keep CBS from airing the segment, BuzzFeed reported.
The White House referred the matter to Trump's personal attorney for comment.
There have been all sorts of complications surrounding this incident-like whether the 2016 payment from Trump's lawyer, Michael Cohen, could have violated election law-but Daniels, meanwhile, has been in the spotlight for weeks.
On Friday, the lawyer filed a document to remove the judge assigned to the case.
"We do not believe it is appropriate for her to preside over this case, which obviously involves the president, in light of this fact", said Michael Avenatti, who represents Daniels.
Mary Hearn, a spokeswoman for the court, said Feffer would not be able to comment because of judicial ethics involving pending cases.
But it is unclear what kind of action could actually win support from a judge - the American legal system nearly never allows such "prior restraint" of speech.