Preet Bharara, the U.S. attorney known for cracking down on insider trading and public corruption in New York, is tackling a new challenge: the growing opioid epidemic.

In a recent initiative, Mr. Bharara has asked local police departments to begin systematically reporting drug overdoses to his office of federal prosecutors in the Southern District of New York, which includes Manhattan, the Bronx and Westchester County. The goal is to treat every overdose like a potential crime scene, including stringent evidence…