Michael Cohen, U.S. President Donald Trump's personal attorney, walks to the Loews Regency hotel on Park Ave on April 13, 2018 in New York City.

Federal prosecutors in a court filing Thursday morning used comments from a new interview by President Donald Trump against his lawyer Michael Cohen to undercut a key claim by Cohen.

Prosecutors also used public statements from Fox News host Sean Hannity to similarly undercut an argument by Cohen that much of the evidence recently seized from Cohen in an FBI raid could be subject to attorney-client privilege.

The shots at Cohen and the president came in a footnote to a filing in Manhattan federal court, where prosecutors announced they were dropping their objection to a special independent watchdog reviewing Cohen's files for such privileged documents.

Prosecutors have previously said they feared Cohen would use claims of privilege to slow down by months, or even years, a review of those documents before they could be used against him in a potential criminal case.

A hearing about the documents is set for Thursday.

Cohen had originally claimed that "thousands if not millions" of pages of documents seized by the FBI on April 9 as part of an ongoing criminal investigation of the lawyer were privileged, and thus barred from being shown to prosecutors probing him.

But in a hearing last week, Cohen's attorneys said he had just three current legal clients, Trump, Hannity and Elliott Broidy, a venture capitalist.

Prosecutors noted on Thursday, that after exposure of his name last week, Hannity claimed, "Michael Cohen has never represented me in any matter."

"I never retained him, received an invoice, or paid legal fees," Hannity said, according to the footnote in Thursday's filing by prosecutors in the United States Attorney's Office for the Southern District of New York.

The prosecutors also pointed out that during an interview with "Fox and Friends" on Thursday, shortly before their filing, Trump had said that "Cohen performs 'a tiny, tiny little fraction' of his overall legal work."

"These two statements by two of Cohen's identified clients suggest that the seized materials are unlikely to contain voluminous privileged documents, further supporting the importance of efficiency here," prosecutors wrote.

That reference to "efficiency" reflects the desire by prosecutors to soon be allowed to sift through Cohen's files for possible evidence of criminal activity.

Cohen's lawyers and lawyers for Trump originally wanted to be the ones who decided initially which documents were privileged and which were not.

The prosecution's decision to endorse the appointment of a "special master" to review those documents for sections subject to attorney-client privilege protection is likely to lead Judge Kimba Wood to name one soon.

Prosecutors had originally asked that a so-called taint team of prosecutors not involved in the investigation of Cohen be the ones to review the documents for privileged files.

On Thursday, CNBC called Cohen for comment on Trump's remarks about him. In addition to saying that Cohen had done just a small fraction of his legal work, Trump had also said that the lawyer had handled "this crazy Stormy Daniels deal."

Daniels is a porn star who has said Cohen paid her $130,000 right before the 2016 presidential election in exchange for her silence about an alleged affair with Trump. The White House has denied such an affair.

Cohen, when asked what he thought about Trump's statements on "Fox and Friends," told CNBC, "I'm on the other line with my lawyers." He then hung up the phone.

There has been widespread speculation that Cohen, who is married and has several children, will agree to cooperate with prosecutors as a part of a deal to receive leniency. Such a deal could include giving prosecutors information about Trump.