Prosecutors said those reviews found "zero" emails between Cohen and Trump and instead indicate that Cohen "is in fact performing little or no legal work" that would be subject to attorney-client privilege.
U.S. District Judge Kimba Wood has scheduled a hearing in the case for Monday afternoon.
The prosecutors assert that they have confidence that any seized material would not fall under the significant amount of attorney-client privilege that Cohen has claimed.
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Wood said she would consider the idea. The hearing followed a raid last week on Cohen's home and office.
Cohen, dressed in a dark suit, white shirt and blue tie and carrying an umbrella, said nothing as he strolled past a throng of at least 100 reporters and photographers and into the Pearl Street courthouse.
That was the revelation in a federal courtroom in New York Monday, where lawyers for the president, the Justice Department and Trump attorney Michael Cohen squared off in in a high-stakes clash over attorney-client privilege.
A source familiar with the raids said last week that the information Federal Bureau of Investigation agents were seeking included information about payments to Daniels.
"The president objects to the government's proposal to use a "taint team" of prosecutors from the very office that is investigating this matter to conduct the privilege review of documents seized from his own lawyer's office", the letter read.
Prior to that, they said, the files seized in the raids might include legal correspondence with five different law firms Cohen had hired on various matters, "hundreds" of clients he had represented dating back to 1991, and the Trump Organization, where he served as legal counsel from 2007 to 2017.
The Southern District of NY prosecutors also heard from Trump Organization lawyers, who want them to throw out all communication between Cohen, the company and its employees.
Prosecutors are reportedly investigating potential campaign finance law violations by Cohen.
Cohen, Trump's lawyer for years, has admitted paying Daniels $130,000 before the 2016 election.
While the WSJ does not mention the facilitation of any Stormy Daniels- or Karen McDougal-style payout, it does quote Us Weekly sources who describe Cohen as "one of these NY characters" whose tirades upon learning of the O'Day story's looming publication were so "totally over-the-top threatening" that they merited the occasional speakerphone mute.
On Monday, Cohen's attorney called the searches "completely inappropriate and unnecessary".
Trump, or his lawyers, would then get to say what he believed to be off-limits to investigators.
The lawyers wrote in a court filing Monday that the raids a week ago to gather evidence from attorney Michael Cohen were "completely unprecedented". Mueller's office referred the case about his business dealings to NY but hasn't been involved since, the Manhattan prosecutors said.
The eight-page letter written by Trump's lawyer, Joanna Hendon, accuses the Justice Department of acting in "an aggressive, intrusive, and unorthodox manner" in an attempt to "eliminate the president's right to a full assertion of every privilege argument available to him".
Both McDougal and Daniels say they had affairs with a married Trump, which the president has denied.