McDougal has said she carried on an affair with Trump in 2006 after the birth of his son. The hearing was set for Monday afternoon.
In Friday's filing, prosecutors said it would be "unprecedented" to allow Cohen's lawyers to decide what it is privileged, and that the government should be allowed to use its own "taint team", or "filter team", to do the job.
An unidentified Trump adviser told the outlet that Cohen, who has worked with Trump for decades, had a "proclivity to make tapes" of private conversations and store them digitally.
Federal prosecutors say in a court filing that the criminal probe that led them to raid the offices of Donald Trump's personal lawyer this week is focused on his "personal business dealings".
Under Justice Department regulations, Mueller must consult with Rosenstein when his investigators uncover new evidence that may fall outside his original mandate.
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But they provided new details on the investigation, which they said has been going on for months.
They also downplayed the scope of potential privilege, saying they had before Monday secretly searched multiple email accounts belonging to Cohen, and which they said indicated that Cohen "is in fact performing little to no legal work".
Cohen has said neither the Trump Organization nor the Trump campaign was involved in the transaction with Daniels and he was not reimbursed for the payment.
Public corruption prosecutors in the USA attorneys office in Manhattan are trying to determine, according to one person familiar with the investigation, if there was any fraud related to payments to Karen McDougal, a former Playmate, and Stephanie Clifford, who performs under the name Stormy Daniels.
The FBI agents who raided the offices of President Donald Trump's personal lawyer, Michael Cohen, sought records of his communications with the Trump campaign as part of what appeared to be an examination of his efforts to quell potentially damaging information about the candidate.
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Before the adjournment, lawyers for Trump and Cohen told a federal judge in NY that they believe some of the documents and devices seized from Cohen during an FBI raid are protected by attorney-client privilege, and they want a chance to review the material before prosecutors get to examine them.
The FBI initiated the Manhattan raid after Mueller, who is authorized to investigate matters related to Russian election meddling, referred information about Cohen to NY authorities.
The warrants also sought business records on Cohen's dealings in the taxi industry, the person said.
Agents also seized records related to a $130,000 payment made to Daniels, who says she had sex with Trump the same year as McDougal. Trump has called the raids a "witch hunt" and "an attack on our country".
A search warrant revealed the tape information was one target of the raid, according to three anonymous sources who spoke with the New York Times. He did not disclose the reason for the court proceeding.
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But a source said it was Cohen's "standard practice" to play recordings for the president that he'd made of other top advisers.