Judge S. James Otero said he will give Cohen until next Wednesday to file a declaration that he will refuse to answer questions in Daniels' suit because of his Fifth Amendment right against self-incrimination.
The judge could rule on the request to put the Daniels case on hold after that.
Cohen is seeking to stay Daniels' suit for up to 90 days because he is facing a federal criminal investigation by prosecutors in New York. Daniels' attorney opposes the delay.
Cohen whose office and home were raided April 9 by FBI agents, wants to avoid making statements in Daniels case that could potentially be used against him in the criminal case, according to Cohen's own attorneys.
Daniels's suit seeks to be released from a non-disclosure agreement she signed barring her from publicly discussing an affair she claims she had with Trump in 2006.
She also claims that Cohen, who paid her $130,000 for the agreement, recently defamed her by implying she is lying.
The potential that Cohen could soon be indicted in New York case was raised several times by his lawyer and Otero at a hearing on the requested delay in Daniels' case in U.S. District Court in Los Angeles.
Neither Cohen nor Daniels, whose real name is Stephanie Clifford, was in the courtroom for the hearing, which lasted slightly longer than one hour, and which was packed with reporters and other spectators.
The non-disclosure agreement with Daniels is being eyed by prosecutors in New York.
A lawyer for Cohen, Brent Blakely, said at the hearing that the "potential indictment of client" puts Cohen "in a very difficult situation."
If Cohen had to testify in the Daniels lawsuit "he would have to plead the Fifth [Amendment] for everything," Blakely said.
But Judge Otero noted that to date the only claim that Cohen would plead the Fifth has been made by Cohen's lawyers.
"Cohen has not filed that his Fifth Amendment rights would be violated without a stay," said Otero at the hearing
"Your burden is to show that there is a large degree of overlap" between the Daniels lawsuit and the criminal investigation, Otero told Cohen's lawyer.
Daniels' attorney, Michael Avenatti, in opposing the delay in the case told Otero that the phrase "justice delayed is justice denied has never been more appropriate" than in this instance.
"They can't continue to hold the agreement over my client's head," Avenatti said. He added that the longer the case drags on, the greater the likelihood that Cohen will be unable to pay any potential judgment in Daniels' favor.
Daniels' lawsuit claims the non-disclosure agreement is invalid because Trump never signed it.
The White House has denied that Trump had an affair with Daniels, and Trump has said he was unaware that Cohen cut a non-disclosure deal with Daniels on his behalf before the election.
Nonetheless, both Trump and Cohen are seeking to have Daniels' lawsuit removed from federal court, where the public can watch arguments play out. Instead, Cohen and Trump want a private arbitrator to rule on the dispute case, as is contemplated by the agreement itself.
Avenatti opposes private arbitration, saying it is not required because the underlying agreement is invalid due to the lack of Trump's signature.
Prosecutors in New York are eyeing Cohen's business dealings in their probe. But they also are interested in the payments to Daniels and in a payment that the publisher of the National Enquirer made to another woman who said she had an affair with Trump. The tabloid newspaper never ran the story by that woman, Playboy Playmate Karen McDougal.