The federal judge overseeing the investigation of Michael D. Cohen, President Trump’s longtime lawyer and fixer, said on Tuesday that a painstaking review of a trove of documents and data files seized from Mr. Cohen in April must be finished by next week.

For nearly two months, lawyers for Mr. Cohen, Mr. Trump and Mr. Trump’s private business, the Trump Organization, have been working with a court-appointed arbiter to determine how much of the seized materials are protected by attorney-client privilege. The review is important because it will shape the contours of the evidence that prosecutors can use as they investigate Mr. Cohen’s business dealings, including hush-money payments he made to women who say they had affairs with Mr. Trump.

Over the weekend, lawyers for the Trump Organization asked the judge, Kimba M. Wood of Federal District Court in Manhattan, to give them until July 11 to finish the review, saying they had just received some new materials, among them “a number of audio files.” Other lawyers in the case have said that Mr. Cohen often recorded his business conversations, but the Trump Organization’s mention of the audio files was the clearest sign yet that Mr. Cohen did in fact make recordings.

Judge Wood was already concerned that the process of sorting through the nearly four million files seized from Mr. Cohen was moving too slowly, and in an order issued on Tuesday, she denied the Trump Organization’s request for two and a half more weeks, instead setting a deadline of 11:59 a.m. on July 5. Any files that the Cohen and Trump legal camps had not deemed to be privileged by the deadline, she said, would be turned over to a special group of prosecutors separate from the investigation. That group, known as a taint team, would then do their own review of the remaining materials to determine which were privileged.