Trump has said he had no knowledge of the payment.
The raid was overseen by the U.S. Attorney's office in Manhattan and was based in part on a referral from special counsel Mueller, said Cohen's lawyer, Stephen Ryan.
The FBI raided Cohen's offices, home and hotel room earlier this week, taking his computers and phones and seizing documents related to a payment he said he'd made to adult film actress Stormy Daniels.
Today, prosecutors acknowledged publicly for the first time Friday that they seized records this week from Cohen.
An attorney for the president, Joanna Hendon, told the judge that Trump has "an acute interest in this matter".
The source says it allows the woman to speak about and take legal action concerning her alleged pregnancy if she so chooses. When asked whether he was concerned about what federal agents might find, Trump said, "No, I'm not". "Well, I think it's a disgrace what's going on - we'll see what happens".
The deal prohibited the Los Angeles woman from disclosing her relationship with Broidy, the Journal said.
Prosecutor Thomas McKay, who agreed to do nothing with the seized materials until Monday's hearing, told Wood that Trump should get no special privileges. The FBI says it has some 7,800 devices inaccessible due to encryption, and Wray has called unbreakable encryption an "urgent public safety issue".
"We heard he had some proclivity to make tapes", said one Trump adviser, who spoke on the condition of anonymity because of the ongoing investigation. Cohen was inside the New York City hotel room where he's been staying when Federal Bureau of Investigation agents suddenly showed up at his door.
Each search warrant specifically describes how records seized that may fall into attorney-client privilege will be filtered out by a team that's walled off from the federal agents who have conducted the investigation, the prosecutors wrote.
Monday's search was broader than previously reported: in addition to Cohen's home and office, there were court-authorized searches of a safety deposit box and two cellphones. 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. He did not disclose the reason for the court proceeding.
A hearing on the searches has been set for 10:30 a.m. (1430 GMT) in a Manhattan federal court, the courthouse said. (He also took the time to complain to the reporters that Attorney General Jeff Sessions had recused himself from the Russian investigation and that Hillary Clinton was not being investigated for "the terrible things that she did and all of the crimes that were committed.") He sounded as if he were trying to build a case for dismissing the special counsel, referring to Mueller and his investigators as "the most biased group of people". Karen McDougal, a former Playboy playmate who said she had an affair with Trump in 2006, said she sold her story to the National Enquirer, which did not publish it. McDougal said in a lawsuit that the Enquirer's parent company, American Media "worked secretly with Mr. Trump's personal 'fixer, ' - Cohen - to suppress her story".
Cohen said he was not reimbursed by the Trump campaign or the Trump Organization, and President Trump said he did not know about the payment or where the money came from. White House attorney Ty Cobb and White House Counsel Donald McGahn have repeatedly sought to dissuade Trump from firing Mueller and renewed their efforts after the raids, two United States officials familiar with the discussions told Reuters.