WASHINGTON—The real-estate company run by the family of White House adviser Jared Kushner in mid-March received a federal grand-jury subpoena for information related to paperwork the company filed in New York City concerning its rent-regulated tenants, according to people familiar with the matter.

The subpoena, part of an investigation by prosecutors in the Brooklyn U.S. attorney’s office, came shortly after an Associated Press article about the company having filed documents that said it had zero rent-regulated tenants,...