Cohen has admitted personally paying the money to Daniels, whose real name is Stephanie Clifford, but has not explained what it was for.
The White House has denied that Trump had sex with Daniels, and Trump has said he was unaware of Cohen's deal with her at the time it was executed.
Daniels' lawyer Michael Avenatti did not immediately return a request for comment from CNBC.
Avenatti is representing Daniels in a lawsuit against Trump that seeks to void the non-disclosure agreement she signed with a company set up by Cohen on the grounds that Trump himself never signed it.
Last month, Avenatti pointedly declined to answer when asked if he had been contacted by the office of special counsel Robert Mueller.
"I'm not at liberty to answer that question one way or the other," Avenatti told CNBC last month
Cohen on Monday told CNBC he would later return calls for comment on his case, but has not done so yet.
The New York Times on Tuesday reported that in addition to seeking information about the payment to Daniels, the search warrants for Cohen's files also were seeking information about a $150,000 payment made to former Playboy model Karen McDougal, who has also claimed she had an affair with Trump in 2006.
The Wall Street Journal reported that prosecutors have also asked for documents related to the payment to Daniels from the Trump Organization, where Cohen is general counsel.
And CNN said that in addition to the payments to Daniels and Douglas, the search warrants for Cohen's files sought records related to his ownership of medallions for New York City taxi cabs.
On Monday, Avenatti told MSNBC: "Mr. Cohen has been placed in the crosshairs by Mr. Trump. And he has been set-up to take the fall."
"In my opinion, an enormous amount of misplaced faith has been placed on his shoulders and I do not believe he has the mettle to withstand it. If I am correct, this could end very, very badly for Mr. Trump and others," Avenatti said.
Mueller is investigating Russian interference in the 2016 election and possibly collusion with Russians by the Trump campaign.
The raids on Cohen's office and residence Monday was spurred by information Mueller's investigation came across, which Mueller referred to the U.S. Attorney's Office for the Southern District of New York.
After news of the raids broke, Trump angrily lashed out at federal investigators during a White House event.
"It's a total witch hunt," Trump fumed.
He called the raids, which were approved by a federal judge and by deputy Attorney General Rod Rosenstein, "an attack on our country in a true sense. It's an attack on what we all stand for."
Asked if he would fire Mueller, Trump replied, "We'll see what happens."
"Many people have said you should fire him."
