A growing number of Democrats have endorsed a single-payer health bill to be proposed Wednesday by Sen. Bernie Sanders, embracing a bigger government role in health care after years of opposing that approach.

Supporters of Mr. Sanders’s bill include several Democratic senators considering a bid for the Oval Office in 2020: Cory Booker of New Jersey, Kamala Harris of California, Kirsten Gillibrand of New York and Elizabeth Warren of Massachusetts.

No major Democratic presidential candidate, other than Mr. Sanders of Vermont, who ran unsuccessfully for the White House last year, has supported a single-payer plan in recent campaigns.

In the House, 117 Democrats are co-sponsoring a plan by Rep. John Conyers (D., Mich.) that would let people of all ages get Medicare—more than double the number who signed onto a similar bill two years ago. In New York and California, single-payer bills passed one chamber of the states’ legislatures this year before stalling.

While more Democrats are signing onto the approach, neither of Congress’s Democratic leaders, Sen. Chuck Schumer of New York nor Rep. Nancy Pelosi of California, have signed onto Mr. Sanders’s bill.