He was ordered to do so by a federal court judge in New York on Monday as adult-film actress Stormy Daniels looked on.

Hannity is a conservative television host known for passionately advocating for Trump on his Fox News show, and often receiving public praise from Trump in return. Calls to a Fox News spokeswoman were not immediately returned.

Cohen, Trump's fiercely loyal and pugnacious lawyer, was in court to ask a judge to limit the ability of federal prosecutors to review documents seized as part of a criminal investigation. The investigation has frustrated the White House as it has spread to enfold some of Trump's closest confidantes.

But in the background, Cohen also had to contend with Daniels' efforts to keep attention on her story, relating to what she says is a past affair with Trump.

Daniels is engaged in a separate civil legal fight over $US130,000 she received in a 2016 agreement arranged by Cohen to stop her from discussing a sexual encounter she says she had with Trump a decade earlier.

Photographers knocked over barricades outside the courthouse as they scrambled to get pictures of Daniels, whose real name is Stephanie Clifford. Inside, she quietly took a seat in the public gallery with her lawyer.

Cohen has argued that some of the documents and data seized in last week's raids are protected by attorney-client privilege or otherwise unconnected to the investigation. But Judge Kimba Wood rejected his efforts to mask the identity of Hannity, a client Cohen had said wanted to avoid publicity.

"I understand if he doesn't want his name out there, but that's not enough under the law," Wood said, before ordering a lawyer for Cohen to disclose the name.

Cohen has asked the court to give his own lawyers the first look at the seized materials so they can identify documents that are protected by attorney-client privilege.

Failing that, they want the court to appoint an independent official known as a special master, a role typically filled by a lawyer, to go through the documents and electronic data seized under a warrant and decide what prosecutors can see.

Prosecutors have asked that the documents be reviewed for attorney-client privilege by a "filter team" of lawyers within their own office, who would be walled off from the main prosecution team.

A lawyer for Trump, Joanna Hendon, asked in a filing on Sunday to be allowed to review documents that in any way relate to the president, which she described as being seized amid a "highly politicised, even fevered, atmosphere." She also appeared in court on Monday.

Australian Associated Press