Stacie Dusetzina, an associate professor of health policy at Vanderbilt University School of Medicine, said a balance would need to be struck between excluding drugs as a way of saving money, and not limiting access to drugs that patients need.

Republicans have proposed block grants in various forms for decades, going back to the Reagan administration. Most recently, Republicans’ bills to repeal and replace the Affordable Care Act in 2017 proposed giving states a choice between a fixed annual sum per Medicaid recipient or a block grant — both of which would have led to major cuts in coverage over time, analysts projected. Concerns from moderate House Republicans about the potential of deep cuts to Medicaid — which now serves more than 71 million people, or more than 1 in 5 Americans — helped doom the repeal effort.

Conservatives say Medicaid spending, which consumes a major and growing portion of the federal and states’ budgets — it cost about $620 billion in 2018, and accounted for 9.5 percent of the federal budget — needs to be reined in. They contend that the current system of unlimited federal matching funds has encouraged states to milk as much as they can, sometimes wastefully. Capping funding, their argument goes, would make Medicaid more efficient and ensure it can continue to help the sickest and most vulnerable Americans.

Although the federal government generally pays between 50 and 77 percent of a state’s total Medicaid costs, depending on the state’s wealth, it covers much more — 90 percent of the costs — for the adults who became eligible for Medicaid under the Affordable Care Act. While most adults who qualified for Medicaid in the past were disabled, pregnant or extremely poor parents of small children, the newly eligible group — nearly 15 million people, Ms. Verma said — includes adults who may be healthy and childless and have somewhat higher incomes.

Ms. Verma has often suggested that adding healthy working-age adults to the program has threatened its viability for more fragile populations, like children, the elderly and the disabled. She did so again on Thursday, saying in a call with reporters that Medicaid “was not originally designed for this group” and that many states had been “far too lax” in verifying whether people are even eligible.

The popularity of Medicaid raises the question of why Mr. Trump would sign off on a contentious block grant program heading into his re-election campaign. The threat posed to the program by Republican efforts to repeal the Affordable Care Act helped Democrats retake the House in the 2018 midterm elections. But if the new option is taken up by states that have not yet expanded Medicaid at all, Mr. Trump could point to it as a new expansion of health coverage.

“Trying to get to the bottom of the politics of this is hard,” said Sara Rosenbaum, a professor of health law and policy at George Washington University. “I assume that they are trying to look like they are helping poor people, without in any way extending the entitlement.”