"The President and Mr. Cohen have purposely ignored our settlement offer, thus doubling down on their efforts to muzzle Ms. Clifford and prevent her from telling the American people what happened", Avenatti tweeted.
The White House has denied Trump had an affair with Daniels.
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.
Cohen has acknowledged making the payment, but has denied the affair took place.
Daniels went on to say that it was "horribly embarrassing" that the first person written on her three-person list was the best in bed of them all, and said the man reached out to her "twice a month".
Cohen declined to comment to CNN on Sunday, deferring questions to his own lawyer.
Avenatti in a letter to Cohen, a copy of which was seen by Reuters, said the actress would wire the funds to an account of Trump's choosing by Friday.
Once a new agreement is signed, the letter said, Clifford would also be able to use and publish any text messages, photos and videos relating to Trump that she might possess.
And it would require the agreement be signed by all parties, including the president, who did not put his name to the previous nondisclosure deal. The White House also did not immediately respond to a request for comment.
Clifford filed a lawsuit in Los Angeles last week that sought to invalidate the agreement so she could "set the record straight".
Clifford could be subjected to a $1 million penalty if she broke the nondisclosure agreement she signed.
CNN's Andrew Kaczynski reported Tuesday that Daniels was asked to write a list of famous men she'd had sexual relationships with when appearing on a Florida radio show hosted by "Bubba the Love Sponge Clem".