Other law enforcement and intelligence agencies, particularly the F.B.I. and C.I.A., have opposed the department’s overseas deployment. Mr. Kelly was criticized for sharing information about terror attacks in London in 2005 and Mumbai in 2008. Federal officials have also complained about the police “freelancing” their own terrorism investigations.

Critics say the Police Department’s liaison program has expanded its focus to justify its ongoing existence without answering deeper questions about the necessity of the initiative.

“It’s a completely wide-open program established with no oversight at all,” said Mike German, a former F.B.I. agent and a fellow at the Brennan Center for Justice.

Critics have also raised questions about the influence of the police foundation’s donors over the department’s policies, and the absence of government oversight for the nonprofit that was founded in 1971. For instance, in 2015, the online news organization The Intercept reported that the United Arab Emirates had donated $1 million to the police foundation to aid unspecified “investigations.” A New York police detective has been stationed with the Abu Dhabi police since 2009.

Susan Birnbaum, the president of the police foundation, said the donation from the Emirati government was benign and was not part of a quid pro quo. She also said the U.A.E. was the only foreign government that had donated money to the organization.

Mr. German, the former F.B.I. agent, said the donors should be disclosed. “What does a foreign government think that it’s getting in exchange for its money?” Mr. German said. “How can New Yorkers be confident that information gathered here in New York City isn’t being shared with foreign governments inappropriately?”

Chief Galati said the department carefully scrutinized each intelligence request from an overseas law enforcement agency to make “certain that it’s legally sound in America.” In the past, the department has rejected inquiries from foreign governments about political dissidents.