ATLANTA >> The cost of expanding Medicaid under President Barack Obama’s health care overhaul is rising faster than expected in many states, causing budget anxieties and political misgivings.

Far more people than projected are signing up under the new, more relaxed eligibility requirements, and their health care costs are running higher than anticipated, in part because the new enrollees are apparently sicker than expected. Rising drug prices may also be a factor.

As a result, at least three expansion states, Arkansas, Kentucky and Ohio, have been pushing to require Medicaid recipients to pay more toward their health insurance — a step that some experts say could lead tens of thousands of poor people who can’t afford the extra cost to drop off the Medicaid rolls and go without coverage. Contrary to common perceptions, Medicaid coverage isn’t always free — many states require modest copayments.

The soaring tab for the state-federal program could also harden opposition to expansion in the 19 states that have yet to sign up. The situation will be one of the thorny issues the next president will have to address, with the health care of millions of Americans hanging in the balance.

Thirty-one states and the District of Columbia have opted to expand the program for poor and lower-income Americans as part of the Affordable Care Act, helping to greatly reduce the number of people without coverage. The federal government has been picking up the entire tab for the new enrollees, but states will begin paying a 5 percent share in January, and that will rise to 10 percent by 2020.

While that may not sound like a lot, it can add up to tens of millions of dollars per state. Health care is already the second-biggest state expense, behind public schools, and a number of states have yet to see their tax revenue fully rebound from the Great Recession.

Arkansas — where more than 307,000 people have signed up for the expansion, exceeding the projected 250,000 — is asking the federal government for permission to charge some new Medicaid recipients premiums to help cover the costs. Some lawmakers want to kill the expansion altogether.

“The facts are, it is what every other (Medicaid) program has been — a fiscal failure,” said state Sen. Bart Hester, a Republican.

In Kentucky, a state struggling to manage its pension debt, Republican Gov. Matt Bevin is asking for federal authorization to charge Medicaid enrollees up to $15 a month. He also wants to require some of them to perform “community engagement,” which would include working or volunteering.

Some 400,000 Kentuckians have enrolled in Medicaid under the expansion, more than double the initial projections, prompting officials to allocate over $257 million for fiscal years 2017 and 2018. That’s well beyond the initial estimate of $107 million.

“At the end of the day, there needs to be the ability to pay for this,” Bevin, who like the state of Arkansas is waiting for an answer from Washington, said last month. “Why trick people, why lie to people, which is what we would be doing if we were to promise something that there was not the ability to pay for?”

Kentucky officials have estimated the proposed changes could result in at least 86,000 people dropping off the Medicaid rolls by 2021. One of them is Steve Olshewsky.

He makes about $9,000 a year teaching online classes and said he could not afford a monthly premium. He has struggled with depression and said he is doing better since he enrolled in Medicaid.

“I’m happy with my life,” he said. “But if you pull this rug out from under me, that’s the end of that story.”

Under the Affordable Care Act, states have the option of expanding Medicaid by loosening the eligibility requirements to include all adults with incomes at or below 138 percent of the federal poverty level, or roughly $16,390 for an individual.

The Associated Press reported last year that more than a dozen states that opted to expand Medicaid had seen enrollments and costs surge beyond their expectations. In a report to Congress over the summer, the federal Centers for Medicare and Medicaid Services said the cost of expansion was $6,366 per person in 2015, about 49 percent higher than anticipated.

Democratic presidential candidate Hillary Clinton has promised that if elected she would work to expand Medicaid in the remaining 19 states. She is proposing an incentive for states that are holding back: three years of full federal financing. Republican Donald Trump has said he would repeal the federal health care law and supports turning Medicaid over to the states and limiting federal financing.

In Ohio, nearly 715,000 people had enrolled under the expansion as of August, way beyond the original estimates of roughly 447,000 by fiscal year 2020. Ohio ended up budgeting more than double the $55.5 million it estimated the new enrollees would cost in the coming fiscal year.

The situation took on more urgency after federal officials last month rejected Ohio’s proposal to charge some recipients as much as $99 a year.

Lawmakers in some states that have refused to expand Medicaid continue debating it, with some opposed for financial reasons, philosophical ones or both.

In Georgia, the state Chamber of Commerce recently produced three options for expansion. Each includes charging premiums.

Republican state Rep. Terry England, chairman of the House budget committee, is among those still dubious about expanding Medicaid. He said the health care system in the U.S. is broken.

“I’m a pretty firm believer that throwing more money at it doesn’t fix it,” England said. “You just prolong the agony until you sit down and fix the problem.”