Federal prosecutors, at the referral of special counsel Robert Mueller's office, executed the raid in NY on Monday.
The Friday motion said that because of that overlap, "Mr Cohen's Fifth Amendment rights may be adversely impacted if this case proceeds", a reference to the possibility that his testimony could be used by prosecutors to build a related criminal case against him.
The prosecutors noted Cohen had personally not turned over any documents to Robert Mueller's Special Counsel investigation. Russian Federation and Trump have both denied allegations of collusion, and the Kremlin has called accusations that it interfered in the election "absurd".
A White House spokesman referred a request for comment to Mr Cohen and his attorney.
Trump has denied any knowledge of Cohen's payment to secure Daniel's October 2016 signature on a secret agreement preventing her from talking about the affair.
According to The Washington Post, Cohen may be under examination for possible bank fraud and for campaign finance violations.
Federal agents have searched through several different emails maintained by Cohen.
Cohen sued in federal court in NY to block prosecutors from accessing files seized from his office and residence in an FBI raid on Monday. On Monday, Trump called the raid an "attack on our country in a true sense".
Trump joined an effort by Cohen to assert attorney-client privilege over records covering Cohen's representation of Trump over the past decade, setting up another extraordinary conflict between the president and his own justice department.
A U.S. District Judge on Friday reluctantly agreed to give attorneys for Michael Cohen an extension to provide evidence bolstering their claim that much of the information seized from Cohen by federal agents Monday is protected by attorney-client privilege. Officials redacted a section in the document explaining why they thought they could not trust Cohen to turn over records willingly.
"Absent a search warrant, these records could have been deleted without record, and without recourse", prosecutors wrote.
Federal investigators specifically asked for all communications about the recording - which surfaced a month before the 2016 presidential election - that personal attorney Michael Cohen had with then-candidate Trump, as well as with his campaign aides. Cohen's lawyer is asking for the lawyers themselves to review the material or a "special master" to decide what is relevant to the case, and Hendon is concerned about a taint team or however the material is handled.
It has been an extremely awful day for President Trump's personal lawyer/fixer/capo. "This is of most concern to him". She said she is considering whether the transcript of that sidebar would be made public, as long as potentially innocent individuals' names are given pseudonyms.
McKay, the assistant USA attorney, told the judge that he believed the proceedings were an attempt to delay the processing of seized material.
Cohen's lawyers, Todd Harrison and Joseph Evans, rejected that argument in a legal memo.