In 1989, New Hampshire’s Republican governor, Judd Gregg, had a gaping budget hole he didn’t know how to fill. His health secretary came up with a solution: a tax maneuver he’d learned through the grapevine that would force Washington to send the state millions in extra Medicaid funds.

It was called a Medicaid provider tax, and New Hampshire was among the first states to try it. New Hampshire taxed its hospitals and returned dollars to them as higher payments for Medicaid patients’ care. On paper, the tax inflated the state’s Medicaid spending, allowing it to collect more matching funds from the federal government.

“It was a way of the state basically gaming the federal government, for lack of a better term,” Mr. Gregg said recently.

What started as creative budgeting in New England has, over four decades, snowballed into a mainstay of financing Medicaid, the insurance program for the poor that covers 72 million Americans. Every state but Alaska has at least one such tax. In some states, provider taxes and related payments bring in more than a third of overall federal funding for the program.

Long after these taxes have become entrenched, congressional Republicans are now considering curtailing or ending them as one way to achieve the steep federal spending reductions proposed in the House budget. If they did, it would save the federal government about $600 billion over the next decade, a large chunk of the $880 billion in cuts that the House committee that oversees Medicaid has been charged with finding.

The change could hit some Republican-led states the hardest, a recent analysis shows, because their Medicaid budgets tend to be more reliant on the medical provider tax strategy.