WASHINGTON >> Mitt Romney’s promise to restore $716 billion that he says President Barack Obama “robbed” from Medicare has some health care experts puzzled, and not just because his running mate, Rep. Paul D. Ryan, R-Wis., included the same savings in his House budgets.

The 2010 health care law cut Medicare reimbursements to hospitals and insurers, not benefits for older Americans, by that amount over the coming decade. But repealing the savings, policy analysts say, would hasten the insolvency of Medicare by eight years — to 2016, the final year of the next presidential term, from 2024.

While Republicans have raised legitimate questions about the long-term feasibility of the reimbursement cuts, analysts say, to restore them in the short term would immediately add hundreds of dollars a year to out-of-pocket Medicare expenses for beneficiaries. That would violate Romney’s vow that neither current beneficiaries nor Americans within 10 years of eligibility would be affected by his proposal to shift Medicare to a voucherlike system in which recipients are given a lump sum to buy coverage from competing insurers.

For those reasons, Henry J. Aaron, an economist and a longtime health policy analyst at the Brookings Institution and the Institute of Medicine, called Romney’s vow to repeal the savings “both puzzling and bogus at the same time.”

Marilyn Moon, vice president and director of the health program at the American Institutes for Research, calculated that restoring the $716 billion in Medicare savings would increase premiums and co-payments for beneficiaries by $342 a year on average over the next decade; in 2022, the average increase would be $577.

Beneficiaries, through their premiums and co-payments, share the cost of Medicare with the government. If Medicare’s costs increase — for instance, by raising payments to health care providers — so, too, do beneficiaries’ contributions.

And those costs would be on top of the costs involved with a full repeal of the health care law, which would eliminate expanded coverage of prescription drugs, free wellness care and preventive checkups.

“One can only wonder what’s going on inside their headquarters in Boston and among their policy people,” said John McDonough, the director of the Center for Public Health Leadership at Harvard. “But there are only two explanations: Either they don’t understand how the program works, which is hard to imagine, or there is some deliberate misrepresentation here because they know how politically potent this charge is.”

The potency of the Republicans’ charge was evident in the 2010 midterm elections, when they accused Obama and congressional Democrats of cutting Medicare by $500 billion to pay for new coverage under the health care law. They went on to recapture control of the House.

Led by Romney, Republicans revived that line of attack for 2012. But Romney cited a higher $716 billion in the same week that he announced his selection of Ryan, who had supported the reductions until joining the Republican ticket. The different dollar figures reflect a shifting time frame: $500 billion represented the projected Medicare savings in the decade after the 2010 health care law was enacted; $716 billion reflects savings from 2013 through 2022.

The Romney campaign adamantly disputes the critics’ assertions.

“The idea that restoring funding to Medicare could somehow hasten its bankruptcy is on its face absurd,” said Andrea Saul, a spokeswoman for the Romney campaign. She added, “Governor Romney’s plan is to repeal Obamacare and replace it with patient-centered reforms that control cost throughout the health care system and extend the solvency of Medicare.”

What Romney proposes to restore to Medicare, however, is not money but additional costs, for higher payments to hospitals, insurers and other care providers. Lobbying groups representing some care providers accepted those reductions during the health care debate, and in exchange they got the law’s mandate for nearly all individuals to have insurance, which meant that providers and insurers would have millions of new paying patients and policyholders.

But Romney’s policy director, Lanhee Chen, argued in a memorandum last weekend that reducing provider payments would backfire and hurt beneficiaries — a concern about the health care law, known as the Affordable Care Act, that is widely shared.

“In the real world,” he wrote, “the result will be fewer providers accepting Medicare payments and worse care for today’s seniors.”

Chen cited recent annual reports from the trustees for the Medicare program and from the chief actuary of the Centers for Medicare and Medicaid Services, Richard S. Foster. The trustees and the actuary each said that the law could well lead to significant savings in the delivery of health care, but both also suggested that over time the scheduled cuts to providers’ payments could prove unrealistic.

“While the Affordable Care Act makes important changes to the Medicare program and substantially improves its financial outlook, there is a strong likelihood that certain of these changes will not be viable in the long range,” Foster wrote.

He added, “The best available evidence indicates that most health care providers cannot improve their productivity to this degree — or even approach such a level — as a result of the labor-intensive nature of these services.”

Romney has been especially critical of the cuts for insurance companies that provide Medicare Advantage, a popular private-policy alternative to Medicare. “This is the president’s plan: $716 billion cut, 4 million people losing Medicare Advantage and 15 percent of hospitals and nursing homes not accepting Medicare patients,” he said in a recent campaign appearance.

But Medicare Advantage, which was created 15 years ago in the hope that private-market competition for beneficiaries would result in lower prices, has consistently cost more than standard Medicare — costs that Medicare beneficiaries must help subsidize through their premiums.

The reductions for Medicare Advantage providers are “a matter of basic fairness because they’ve been overpaid for years,” Moon said. As for beneficiaries, she added, “they’re guaranteed basic Medicare benefits. They may lose some extra benefits they may have been getting, but in effect you’re saying some of the windfall benefits may go away.”

“The bottom line,” said Rep. Chris Van Hollen of Maryland, the senior Democrat on the House Budget Committee, which Ryan leads, “is that Romney is proposing to take more money from seniors in higher premiums and co-pays and hand it over to private insurance companies and other providers in the Medicare system.”