Citing the Fifth Amendment in the Clifford case allows Mr. Cohen to avoid being deposed and revealing sensitive information in the more important criminal investigation. That investigation — which prosecutors say has been going on for months — became public in dramatic fashion on April 9, when agents from the New York office of the Federal Bureau of Investigation raided Mr. Cohen’s office, apartment and a room at the Loews Regency Hotel he had been using. The inquiry is said to be focusing on hush-money payments that Mr. Cohen made to — or helped arrange for — Ms. Clifford and Karen McDougal, a former Playboy model who has also said she had an affair with Mr. Trump.

For days now, prosecutors from the United States attorney’s office in Manhattan have been sparring with Mr. Cohen’s lawyers — and with lawyers for Mr. Trump — for the right to review the records first, a step that will shape the contours of how the government presses its investigation into whether Mr. Cohen tried to suppress negative news coverage of the president in the run-up to the 2016 election.

Lawyers for Mr. Cohen suggested on Wednesday that they were already taking steps to gird themselves for battle. In new court papers, they said that they were setting up “war rooms” to review the seized materials, testing and vetting “systems” to process them, and marshaling what seems to be a small legal army of “operations and technology personnel, paralegals, discovery experts, associates and senior reviewing partners.”

The martial tone of Mr. Cohen’s filing was in contrast to a similar letter submitted to the court by lawyers for the Trump Organization, who merely noted that they were “fully capable of processing and reviewing the materials.” In a third submission, Mr. Trump’s lawyers said that they would “consult closely with Mr. Cohen’s counsel to assist in identifying the seized materials that relate to the president.”

All of the papers were requested in advance by Judge Wood, who, in a hearing on April 16, put off a decision on how to resolve the conflict over the attorney-client privilege review, saying that she might appoint a special master to oversee the process. Two days later, the prosecutors and Mr. Cohen’s lawyers submitted to Judge Wood lists of potential candidates.