Cohen is being represented by his personal lawyer, David Schwartz, who returned fire at Avenatti last week in a radio interview, calling the attorney a "PR machine" and the case "a complete bunch of nonsense".
Ms Daniels has also told U.S. media that she was threatened by a stranger in a auto park in 2011 to "leave Trump alone".
Daniels, whose real name is Stephanie Clifford, said she received threats to her physical safety if she spoke out, and that she received a $130,000 payment to keep quiet just before the 2016 presidential election.
Another reporter then asked the President: "Do you know where he got the money to make that payment?"
It was the president's first acknowledgment of the scandal that has haunted him for weeks.
Daniels and her lawyer say the agreement is invalid because Trump never signed it.
However she lost a court motion for Mr Trump to give sworn testimony about her claim that they had a relationship.
That claim was immediately challenged by Daniels' lawyer Michael Avenatti.
She's one of at least 19 women who have accused Trump of sexual misconduct - all dating back before the 2016 election.