Federal prosecutors and Department of Transportation officials are scrutinizing the development of Boeing Co.’s 737 MAX jetliners, according to people familiar with the matter, unusual inquiries that come amid probes of regulators’ safety approvals of the new plane.

A grand jury in Washington, D.C., issued a broad subpoena dated March 11 to at least one person involved in the 737 MAX’s development, seeking related documents, including correspondence, emails and other messages, one of these people said. The subpoena, with a...