Avenatti also said he would be demanding access to any audiotapes featuring Stormy Daniels's former lawyer and Cohen.
Both Giuliani and Avenatti have been frequent guests on cable news shows, turning the legal battle swirling around Cohen and Trump into a veritable war of words.
"Any media report citing 'unnamed sources' (and not a single document) suggesting we are delaying the investigation into Mr. Cohen and DJT is completely false and without basis. We have already waived the privilege as to a host of docs and communications to ensure justice is done."
The lawyer mocked Rudy Giuliani and Donald Trump in a Sunday interview.
"Ms. Daniels's case has never been stronger," Michael Avenatti told Newsweek.
"So if I were you, I would tell Mr. Trump to find someone else to fabricate things about me," Avenatti wrote in an email.
"She did not handle the situation appropriately, respectfully or professionally."
Avenatti's critics often point to his media appearances in an attempt to discredit him.
Democrats gush "over the porn lawyer, the one whose eyes are too close together," Carlson said. Michael Avenatti soon fired back.
Michael Avenatti was responding to suggestions that he was a campaign operative.
"You've got millions of dollars that are being deposited into this account," Avenatti said.
"Who knew Mr. Cohen was such a brilliant renaissance man?" Michael Avenatti tweeted.
The lawyer for Stormy Daniels has claimed that $500,000 was deposited into the same account Michael Cohen used to silence Stormy Daniels.
"One of the worst T.V. appearances by any attorney on behalf of a client in modern times."
"For over 9 yrs after the end of the affair ... Mr. Trump had a family. And yet no payment was ever made to 'protect them,'" Avenatti said.
Avenatti made the claim on MSNBC's The Last Word.
One reporter asked "why the president was not truthful with the American people."
The lawyer for Stormy Daniels said he believed intercepted communications led investigators to believe Cohen might destroy evidence.
Conservative commentators and longtime Trump defenders have questioned the president's honesty in the wake of the Giuliani confession.
The Fox News host said she "put her legal hat on" while trying to make sense of the situation.