Earlier Monday, Avenatti, a California lawyer, asked a judge to let him represent his client, whose real name is Stephanie Clifford, in litigation over Cohen's records in NY federal court.
Cohen received additional payments from companies including AT&T, Swiss drug-maker Novartis AG and a US private-equity firm tied to a Russian oligarch.
Minutes after the screenshots containing McNally's numbers were published, Avenatti deleted them and reposted the same tweet with the numbers blacked out.
"Given the dearth of legal or factual support relating to anything having to do with Mr. Avenatti's right to advocate on behalf of his client in this court, Mr. Cohen's argument's must be summarily rejected", the memo stated.
"Avenatti is clearly protected by the First Amendment rights of free speech to publish information on matters that, without serious dispute, are a matter of public concern", according to the filing.
This isn't the first time Avenatti has requested to be admitted to practice in the district, the filing noted.
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In Monday's filing, Avenatti argued that much of the contents of his document has since been proven true. "As discussed in her motion to intervene, Ms. Clifford [Daniels' given name] has very important and legitimate interests in protecting her records".
Michael Avenatti, attorney for Stormy Daniels, leaves the Daniel P. Moynihan Courthouse in Manhattan after a hearing in front of Judge Kimba Wood regarding a search warrant that was executed at the home, hotel and office of Trump's lawyer Michael Cohen. Daniels says Trump and Cohen tried to buy her silence about an affair she claims she had with Trump years ago, shortly after Trump married current wife Melania Trump.
"It doesn't seem like it furthers the legal interests of Stormy Daniels and her desire to be released from the NDA and/or obtain damages from Cohen or the President", Michael Zeldin, a former federal prosecutor who worked alongside Special Counsel Robert Mueller at the Justice Department, told TPM in a phone interview.
Daniels sued Cohen and Trump to get out of the agreement, and on defamation claims over various things both have said about her and the hush agreement in the press and on Twitter.
Wood could be "looking at Avenatti's tendency to be on television and now to release information", Zeldin said. Her case, recently removed to Los Angeles federal court, has been stayed pending the criminal probe of Cohen.