WASHINGTON—President Trump called on his attorney general to sue opioid makers, the latest move by his administration to combat the highly addictive painkillers linked to tens of thousands of U.S. deaths a year.

In a cabinet meeting Thursday at the White House, Mr. Trump asked Attorney General Jeff Sessions to bring a “major lawsuit” against drug companies that “are really sending opioids at a level that it shouldn’t be happening.” He said he wanted the attorney general to bring the suit at the federal level, rather than join...