Washington ? They’re not buying it. Most Americans say they don’t believe Medicare has to be cut to balance the federal budget, and ditto for Social Security, a new poll shows.

The Associated Press-GfK poll suggests that arguments for overhauling the massive benefit programs to pare government debt have failed to sway the public. The debate is unlikely to be resolved before next year’s elections for president and Congress.

Americans worry about the future of the retirement safety net, the poll found, and 3 out of 5 say the two programs are vital to their basic financial security as they age. That helps explain why the Republican Medicare privatization plan flopped, and why President Barack Obama’s Medicare cuts to finance his health care law contributed to Democrats losing control of the House in last year’s elections.

Medicare seems to be turning into the new third rail of politics.

“I’m pretty confident Medicare will be there, because there would be a rebellion among voters,” said Nicholas Read, 67, a retired teacher who lives near Buffalo, N.Y. “Republicans only got a hint of that this year. They got burned. They touched the hot stove.”

Combined, Social Security and Medicare account for about a third of government spending, a share that will only grow. Economic experts say the cost of retirement programs for an aging society is the most serious budget problem facing the nation. The trustees who oversee Social Security and Medicare recently warned the programs are “not sustainable” over the long run under current financing.

Nearly every solution for Social Security is politically toxic because the choices involve cutting benefits or raising taxes. Medicare is even harder to fix because the cost of modern medicine is going up faster than the overall cost of living, outpacing economic growth as well as tax revenues.

“Medicare is an incredibly complex area,” said former Sen. Judd Gregg, R-N.H., who used to chair the Budget Committee. “It’s a matrix that is almost incomprehensible. Unlike Social Security, which has four or five moving parts, Medicare has hundreds of thousands. There is no single approach to Medicare, whereas with Social Security everyone knows where the problem is.”

That’s not what the public sees, however.

“It’s more a matter of bungling, and lack of oversight, and waste and fraud, and padding of the bureaucracy,” said Carolyn Rodgers, who lives near Memphis, Tenn., and is still working as a legal assistant at 74. “There is no reason why even Medicare, if it had been handled right, couldn’t have been solvent.”

In the poll, 54 percent said it’s possible to balance the budget without cutting spending for Medicare, and 59 percent said the same about Social Security.

Taking both programs together, 48 percent said the government could balance the budget without cutting either one. Democrats and political independents were far more likely than Republicans to say that neither program will have to be cut.