WASHINGTON — New momentum for a lasting fix to Medicare’s doctor-payment problem shows that lawmakers may finally be recognizing what has long bothered their constituents.

Year after year, the threat of 20 percent payment cuts averted at the last minute has seemed a curious way to run a program that lives depend on.

“A lot of these doctors are talking about not performing the service,” said Jesse Thomas of Mount Vernon, N.Y., a Medicare beneficiary retired from a career on Wall Street. “They are not in business to do business for a deficit. They may be good guys, but they’re not that good.”

Congress always likes to be in control of spending, added Thomas, but lawmakers need to take better care of beneficiaries who’ve supported Medicare with their tax dollars.

The root of Medicare’s problem is a cost-containment formula included in a 1997 deficit-reduction law. Under Medicare, outpatient care is financed 25 percent with beneficiary premiums and 75 percent from the federal treasury.

Concern that rising health care costs would overwhelm the budget prompted Congress to set a target for outpatient spending, enforceable with payment cuts to doctors. It became a symbol of government dysfunction after lawmakers got into the habit of regularly issuing temporary reprieves, lest doctors stop accepting Medicare patients. But it also just pushed the cuts into the future and let them pile up.

“Everybody hates it,” said Rep. Jim McDermott, D-Wash., who is a psychiatrist. “In the political process there is a time when something gets to where it’s like a boil and it’s about to burst. And then it’s gone.”

The House is expected to vote Thursday on replacing the current formula with a new approach. It calls for a period of basically stable reimbursements, followed by gradually shifting a larger share of doctors’ pay so that it’s keyed to quality, rather than quantity, of service. The Medicare fix is packaged with an extension of children’s health insurance, funding for community health centers and dozens of other provisions. The outlook in the Senate is unclear.

“A permanent fix, rather than another patch, should take away the perennial fear that doctors will stop seeing Medicare patients ... and provide a measure of stability for the program,” said Tricia Neuman, a Medicare expert with the nonpartisan Kaiser Family Foundation.

The fear that doctors would stop treating Medicare patients because of constant uncertainty about their fees never materialized on any large scale. But it’s certainly been part of the conversation.

Dr. Robert Wergin, president of the American Academy of Family Physicians, practices in the small town of Milford, Neb. He remembers when Congress hesitated over the “doc fix,” as it’s known in lawmakers’ shorthand, and as a result Medicare held up payments to physicians. That caused a cash crunch.