Cohen's home, office, hotel room and safety-deposit box were raided by the FBI on April 9 as part of a long-running criminal investigation of his activities. They want the separate group of prosecutors, or a so-called taint team, to review the seized documents to weed out any that might be covered by the privilege.
The hearing in NY was ongoing Monday afternoon.
Now, the president's legal team has made what's considered an extraordinary request that may delay the Cohen investigation.
A person familiar with the raids said last week that the information Federal Bureau of Investigation agents were seeking included information about payments to Daniels.
Trump said Sunday that all lawyers are now "deflated and concerned" by the FBI raid on Cohen, who is under criminal investigation for personal business dealings.
"Michael Cohen has never represented me in any matter".
The government can't look at the substance of the materials, beyond the narrow search.
Cohen is asking for the review to be conducted by a court-appointed attorney. Wood, the Senior United States District Judge for Manhattan's Southern District, was herself no stranger to the harsh glare of the media. She said she trusted independent federal attorneys to sift through the material and determine what is admissible in court. But, as the news of Hannity's role ping-ponged around media and political circles in NY and Washington, Hannity downplayed any attorney-client relationship he has had with Cohen.
On Friday, lawyers for Cohen appeared in federal court in NY asking that they, not the Department of Justice, be given a first crack at reviewing the seized evidence to see if it was relevant to the investigation or could be forwarded to criminal investigators without jeopardizing attorney-client privilege. But she also said she wasn't ready yet to rule on who would eventually get to decide what materials might stay confidential. "That ends now", she added.
Wood also asked both sides to each provide four nominees for special master.
Cohen, who was forced to reveal that Fox News host Sean Hannity is another of his high-profile clients, is being investigated for fraud. Gasps could be heard echoing throughout Wood's stately, large courtroom.
The first issue to be dispensed with was the name of Michael Cohen's super secret client-eventually and painstakingly revealed to be Sean Hannity. Investigators are likely to examine whether the payment was legal under election campaign finance laws and whether Cohen disclosed the true reason for borrowing the funds.
The Hill notes such a move is established procedure meant to "protect the attorney-client privilege and to ensure that the investigation is not compromised", according to the U.S. Attorney's Manual. In other words, the court would have known his identity, but the public would not have.
Fox News host Sean Hannity is seen in the White House briefing room in Washington, D.C., on January 24, 2017. After pausing for a few seconds, Wood finally said, "That's not enough under the law".
Judge refuses privilege review bid by Cohen...
She accuses the Justice Department of acting in an "aggressive, intrusive, and unorthodox manner", and eliminating Trump's "right to a full assertion of every privilege argument available to him".
"It nearly goes without saying, unfortunately, that none of Mr. Cohen's clients want to be associated with the government raid on his home and law office, or want to be affiliated in any way with the proceedings here and the attendant media coverage".
The main point, McKay said, was that the opposing counsel would use a broad claim of privilege to "ask for an inch and take a mile".