It also provides insurance for poor adult Americans, and recent evidence shows that its expansion under Obamacare has given more poor people access to health care services and reduced their exposure to financial shocks.

The Republican approach would set a formula for determining a maximum payment for each person in the program. Then that cap would grow by a set rate each year. Lawmakers are negotiating about the rate to use, but all of the options are intended to grow more slowly than expected under the current system. The gap would be left for states to fill — or cut.

“While details remain elusive, this is shaping up to be the largest intergovernmental transfer of financial risk in our country’s history,” said Matt Salo, the executive director of the National Association of Medicaid Directors, in an email. Mr. Salo said that some of his directors would welcome caps if they came with more program flexibility, but said the current approach amounted to a funding cut.

The growth in medical spending tends to be uneven year over year, which means states might hit the caps in one year and fall under them in another, even without any program changes. Researchers at the Brookings Institution recently looked back at historical Medicaid spending to see what would have happened under a cap. They found that random variation was substantial.

Medicaid advocates worry particularly that a fixed growth rate doesn’t account for this varying pattern of health expenditures, which might shoot up in a year where there’s an epidemic or an important new treatment. Many Medicaid budgets increased in recent years after the introduction of expensive but effective medications for hepatitis C, for example. States had to pay more for the drug, but federal spending also increased to match it.

“Could you imagine tomorrow if finally we had a Zika virus vaccine, and that vaccine costs $50K a dose?” said Sara Rosenbaum, a professor of health law and policy at George Washington University. “Would you not want every woman of childbearing age to be immunized?”

Advocates for the structural change point to inefficiencies and waste in the current program. There is some evidence that Medicaid programs enroll some people who are not eligible and sometimes cover some services that are not medically necessary. James Capretta, a fellow at the conservative American Enterprise Institute, said that the current system, where the federal government matches all state spending, discourages efficiency.