Cohen has denied Daniels' claims but admitted to making the payment. The additional claim that Daniels filed Monday states that Cohen's denial is a defamatory statement.
"It was reasonably understood Mr. Cohen meant to convey that Ms. (Stephanie) Clifford is a liar," the complaint reads, referencing Daniels' real name.
Avenatti added, "Mr. Cohen made the statement knowing it was false or had serious doubts about the truth of the statements."
Reacting to the latest development, CNN legal analyst Jeffrey Toobin said it is another attempt to compel Cohen, or perhaps even the President, into a deposition under oath.
Toobin said key questions would include whether Cohen actually used his own money to pay Daniels and whether Trump knew about it.