We do not know, of course, what work Cohen completed on behalf of Sean Hannity. Ted Cruz, R-Texas, in Phoenix on March 18, 2016.
Trump's attorneys argue the president and Cohen need to look over those materials first to determine which ones should be excluded under attorney-client privilege.
Daniels' attorney, Michael Avenatti, said Sunday that his client will attend Monday's hearing.
One was Trump himself.
Broidy, a major Republican donor, resigned as fundraising chair for the Republican National Committee last week after it emerged he had paid $1.6 million to a Playboy model who became pregnant during an affair.
The government has asked for the names of all of Cohen's clients, so that it can run searches of the seized material and make an initial determination of what might be privileged. The hearing followed a raid last week on Cohen's home and office.
Wood, though, demanded the name.
Wood said she would consider the idea.
An email sent to Fox News seeking comment from Hannity was not immediately returned.
An exasperated Hannity addressed the revelation on his radio show Monday afternoon, calling it much ado about nothing.
It began with an appearance by porn actress Stormy Daniels, who was swarmed by photographers and almost fell as she was hustled into the courthouse, a scene that captured the sensational atmosphere around a legal fight involving the president and an FBI investigation into his personal attorney.
Last week, a source familiar with the raids said that Federal Bureau of Investigation agents were looking for information on payments to Daniels among materials related to other business dealings. They have agreed to delay the start of the review until the issue is resolved.
The attorneys said that if a special master was appointed, the client would allow their name to be disclosed to that special master. But they also want DOJ and the FBI to investigate whether the representations the Federal Bureau of Investigation made to the secret Foreign Intelligence Surveillance Act (FISA) court to obtain a warrant that started the surveillance of the Trump transition team "were in error". While Cohen is not in the same situation, there are certainly extraordinary circumstances, given that one of his clients in the President of the United States, and there may be documents involving Trump that have nothing to do with criminal activity.
Lawyers for President Donald Trump's personal attorney say investigators "took everything" during raids last week on his residence and office.
Cohen arranged for a payment of $130,000 to Daniels in what Avenatti said was in exchange for a nondisclosure agreement related to a sexual encounter with Trump before the 2016 election.
Federal prosecutors revealed Friday said the criminal investigation had been going on for months.
Prosecutors say that material should be reviewed by a team of Justice Department lawyers independent from the investigation who could identify records that should remain confidential.
Then, the president wants the court to direct Cohen "to identify to the president all seized materials that relate to him in any way and to provide a copy of those materials to him and his counsel", according to the letter.
Stormy Daniels's attorney asked to join in on the hearing, as he said he believes items seized were in regards to his client.
Federal Bureau of Investigation agents raided Cohen's home, office, safe deposit box and cellphones last week.
The attorneys said in the filing that Cohen represented three clients in the past year. 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.