WASHINGTON—The Trump administration is sounding out the medical industry on requiring hospitals, doctors and other health-care providers to publicly disclose the secretly negotiated prices they charge insurance companies for services, a move that would expose for the first time the actual cost of care.

Mandating public disclosure of the rates would upend a longstanding industry practice and put more decision-making power in the hands of patients. Hospitals and insurers typically treat specific prices for medical services as...