Several states are probing pharmacy-benefit managers’ business practices in government-sponsored health programs, adding to the scrutiny the middlemen face in Washington for their role in the cost of drugs.

At least three state attorneys general are investigating PBMs, in addition to other state probes looking into how the companies contract with Medicaid and other programs. CVS Health Corp., Cigna Corp.’s Express Scripts unit and UnitedHealth Group Inc., which dominate the PBM market, are scheduled to appear before the Senate Finance Committee in Washington April 9 to discuss their role as the nation grapples with soaring drug prices.

“I am concerned...