The White House is reviewing a Medicare proposal that could extend the time health-care providers can stay in the no-risk stage of the agency’s coordinated care model.

The stage, known as Track 1 of the Medicare Shared Savings Program, is designed to help accountable care organizations make the transition from traditional fee-for-service Medicare to value-based service aimed at delivering higher quality care at lower cost.

ACOs are groups of doctors, hospitals, and other health-care providers, who come together voluntarily to...