The hearing concerned a motion by Cohen seeking an order restraining federal prosecutors from using materials seized in Monday's FBI raid of his office, home and hotel room.
"Mr. Trump, Mr. Cohen and their teams were still scrambling on Friday to assess the damage from the raid early Monday morning", The Times reported.
But they provided new details on the investigation, which they said has been going on for months. Those emails, they said, indicated that Cohen was "performing little to no legal work, and that zero emails were exchanged with President Trump".
Assistant U.S. Attorney Thomas McKay said the legal issues at stake were no different than other cases involving lawyers' records.
They said agents had already searched multiple email accounts maintained by Cohen.
The raids were conducted after federal prosecutors in Manhattan obtained a search warrant based, in part, on a referral from special counsel Robert Mueller, who is conducting the federal probe into ties between Trump campaign associates and Russian Federation. Initially, Mueller had requested some records from Cohen while he was with the Trump family company - a position he held for about a decade - yet dropped the request after Cohen pushed back.
"Defendants will be substantially prejudiced if compelled to proceed with this case while the criminal investigation related to Mr. Cohen is ongoing, given the substantial overlap between the facts in this action and the criminal investigation, and Mr. Cohen's status as is a key witness." states the filing by attorneys Brent Blakely and Charles Harder.More news: Russia pledged to counter any US missile strike on Syria
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Among the material sought by investigators in the search were records related to payments made to adult-film star Stormy Daniels and former Playboy model Karen McDougal, two women who allegedly had sexual liaisons with Trump years ago.
But when the judge demanded that Cohen's lawyers provide some proof that Cohen has any other clients beyond the commander in chief, they twice came back empty-handed.
Searches of lawyers' offices are rare but not unheard of, and prosecutors use what's called a separate "taint team" to review the material and exclude any documents that are covered by the attorney-client privilege. Afterwards, Avenatti said at a press conference that he wasn't surprised at Trump's interest in the case because longtime lawyer Cohen knew "where all the bodies were buried".
Cohen's reply is due by 6 p.m. Tuesday.
The raid has shaken Trump, who called Cohen on Friday, The New York Times reported.
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".
Joanna Hendon, a new lawyer for Trump who appeared in court, said her law firm, Spears & Imes LLP, was engaged by Trump on Wednesday and she was "not prepared" to present her argument.