The White House has consistently said Trump denies either affair.
Prosecutors say Cohen barely practiced law so most of the documents aren't off limits.
"You could imagine a search warrant that describes that Michael Cohen got a loan from a bank to make a payment on Trump's behalf and made misrepresentations to the bank about how the money would be used", former federal prosecutor Renato Mariotti told Bloomberg.
"It resulted in the unnecessary seizure of protected attorney client communications between a lawyer and his client", attorney Stephen Ryan said in the statement.
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".
Allies of President Donald Trump are reportedly anxious that Federal Bureau of Investigation agents seized taped conversations made by the president's personal lawyer, Michael Cohen, in raids of his office, home and hotel room on Monday.More news: NBA Draft: Sacramento Kings lose tiebreaker for sixth
Public corruption prosecutors in the USA attorney's 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 document was filed publicly after lawyers for Cohen appeared before U.S. District Judge Kimba M. Wood to ask that they, not Department of Justice lawyers, be given the first crack at reviewing the seized evidence to see whether it was relevant to the investigation or could be forwarded to criminal investigators without jeopardizing attorney-client privilege. Cohen's attorney was prepared to resume this afternoon.
Trump allies reportedly fear Federal Bureau of Investigation seized taped...
At a hearing in the U.S. District Court in Manhattan, Joanna Hendon, a lawyer for Trump, said the president had "acute" and "weighty" interests at stake.
Cohen is seeking a temporary restraining order regarding the searches of his home and office, a Justice Department spokesman said earlier on Friday.
In a footnote, prosecutors wrote that although the investigation was referred to prosecutors by Mueller, it was proceeding independently. It's unclear if he ever recorded conversations he had with Trump, but one person said Trump knew Cohen taped people because he'd play the recordings to him.
Federal agents raided Cohen's Manhattan office on Monday, seizing records on topics including a $130,000 payment made to porn actress Stormy Daniels.