Avenatti spoke outside the court at Friday's hearing and then followed with a suggestive tweet that "the weather forecast for Monday looks very Stormy".
Todd Harrison, an attorney for Cohen, urged the judge to issue a temporary restraining order to let either Cohen's lawyers or a court-appointed special master review the materials seized by Federal Bureau of Investigation agents.
Cohen's explanation for a $130,000 hush payment he made just before the 2016 election to Stormy Daniels, an adult film actress who claims to have had an affair with Trump, is that he made the payment by himself, without informing Trump and using exclusively his own personal funds.
Prosecutor Thomas McKay, who agreed to do nothing with the seized materials until Monday's hearing, told Wood that Trump should get no special privileges.
The request is reportedly based on the recent raid of the home and hotel room of Trump's attorney, Michael Cohen. The hearing was set for Monday afternoon.
Wood said the intervention of a court or special master in the process was not unprecedented, citing the prosecution of Lynne Stewart, a radical Manhattan lawyer who was charged with smuggling communications out of prison for one of her clients.
McClatchy's sources did not say whether Cohen took a commercial flight or private jet to Europe, and gave no explanation as to why no record of such a trip has surfaced.
Avenatti told reporters after the Friday morning hearing that he had no doubt the materials seized include documents concerning the negotiations with Daniels, which is the subject of a lawsuit filed by her against Cohen and Trump in Los Angeles.
"As a privilege holder, he [Trump] has an acute interest in those proceedings and the manner in which these materials are reviewed", Hendon said, asking Judge Kimba Wood at the US District Court for the Southern District of NY to postpone the hearing until next Monday.
But they provided new details on the investigation, which they said has been going on for months. Authorities revealed that they searched a number of email accounts used by Cohen.
McKay said Trump's ability to invoke the privilege is "no different" from anyone else's.
In response to Cohen's motion to prevent prosecutors from using evidence collected in Monday's raids of his home and office, the United States attorney in NY asserted the raids were authorized by a federal judge to seek evidence of conduct "for which Cohen is under criminal investigation".
An attorney for the president, Joanna Hendon, told the judge that Trump has "an acute interest in this matter".
President Donald Trump's personal attorney, Michael Cohen, has been under criminal investigation for months by federal prosecutors who empaneled a grand jury to probe his business dealings beyond his law practice, according to a new court filing. Cohen reportedly said he "would rather jump out of a building" than testify against Trump.
Investigators raided Cohen's home and office on Monday.
It isn't clear yet, however, whether the business dealings described in court documents on Friday might have any connection to Mueller's investigation into whether any Americans conspired with Russia's attack on the 2016 presidential election.
It's unclear whether Mueller's investigators also have evidence that Cohen actually met with a prominent Russian - purportedly Konstantin Kosachev, an ally of Russian President Vladimir Putin - in the Czech capital.