In an order issued on Friday, U.S. District Judge Kimba Wood in Manhattan rejected efforts by Trump, the Trump Organization and Cohen to file their objections entirely under seal.
She agreed with the government that the filings should be public except as to portions that "divulge the substance of the contested documents."
Wood said she would decide later which portions could be sealed.
Joanna Hendon, a lawyer for Trump, did not immediately respond to requests for comment. Todd Harrison, a lawyer for Cohen, did not immediately respond to similar requests.
The criminal probe into Cohen's business dealings stems in part from a referral by Special Counsel Robert Mueller, who is investigating whether Trump's 2016 campaign colluded with Russia to influence that year's U.S. presidential election.
The special master, former federal judge Barbara Jones, is reviewing materials seized in April raids of Cohen's home, office and hotel room, to determine which are subject to attorney-client privilege.
On Monday, she said in a report that 162 files, out of more than 292,000 reviewed so far, were privileged or partially privileged, and seven were "highly personal."
Roughly 3.7 million files were seized, Cohen's lawyers have said. Jones is reviewing only those that lawyers for Trump, the Trump Organization and Cohen believe might be privileged.
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