In his filing Monday, Avenatti said that nearly all of the information that was released about Cohen ended up being confirmed either by the very companies involved or through public reporting by news outlets.
However her lawyer Michael Avenatti said on Twitter that Mr Trump's comments on Air Force One had strengthened Ms Daniels's case "exponentially".
Minutes after the screenshots containing McNally's numbers were published, Avenatti deleted them and reposted the same tweet with the numbers blacked out.
Cohen's attorney, Stephen Ryan, argued in court filings last week that Avenatti should be barred from intervening for releasing the document on Cohen's finances.
In his response, Avenatti noted he's been admitted to practice in the Southern District of NY and the Second Circuit previously, and Cohen failed to mention any legal reasons why Avenatti shouldn't be admitted this time.
"That Mr. Cohen may be dismayed that these damaging revelations have come to light and have been proven true does not come remotely close to justifying a denial of Mr. Avenatti's right to appear before this court", the filing Monday stated.
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Avenatti responded in a Monday filing that it was his First Amendment right to publish information about Cohen that is "of the utmost public concern", and that Daniels should be allowed to have counsel fight for her "important and legitimate interests in protecting her records".
Michael Avenatti, the lawyer representing adult film actress Stormy Daniels, defended his explosive disclosure last week of more than $1 million in payments from companies including AT&T Inc.to Michael Cohen, Donald Trump's longtime lawyer and fixer.
The firm, Essential Consultants LLC, is the same one Cohen used to pay US$130,000 to Daniels, as part of an agreement to keep her from talking publicly about her claim that she had sex with Trump in 2006.
The FBI raided his office and home last month, reportedly seizing financial records and documents related to his payment of Daniels and another woman who alleged she had an affair with Trump.
Avenatti also went on the attack in his filing, saying that Cohen's objection to his pro hac vice application is nothing "but a highly improper attempt to soil Mr. Avenatti and unnecessarily lure and entangle this Court into Mr. Cohen's elaborate campaign to now discredit Mr. Avenatti".
Avenatti said Cohen's belief that Avenatti broke the law by publishing Cohen's banking information is misguided.