It was perhaps inevitable that the Republican effort to pass a vast fiscal package this year would, at some point, get caught up in the thicket of the state and local tax deduction.

After all, the deduction, often called SALT, has long had the potential to cause a political standoff. Many G.O.P. lawmakers abhor it and, in 2017, imposed a $10,000 limit on the amount of state and local taxes Americans can write off on their federal returns. But to pass a tax bill this year, the party will need the support of a motivated clutch of Republicans who have made lifting that cap the animating promise of their political careers.

Those lawmakers, who represent high-tax states like New York and New Jersey where the deduction is cherished, say they are willing to tank the package over the issue. Representative Nick LaLota, Republican of New York, can already visualize voting against the bill.

“There’s a green ‘yes’ button and there’s a red ‘no’ button to press. Come time, if there’s not enough SALT in this bill, I’m pressing the red ‘no’ button,” he said. “It is a hill I am willing to stake my entire congressional career on.”

Attempts by House Republican leaders to reach a deal with members like Mr. LaLota yielded little progress this week, leaving the issue unresolved as G.O.P. lawmakers prepare to release the first draft of their tax bill next week. Along with Medicaid, the health care program for the poor that Republicans have targeted for cuts, the state and local tax deduction could determine the fate of the entire G.O.P. legislative agenda.

That’s because any change to the current $10,000 limit would be incredibly expensive, threatening to swamp the overall Republican budget for tax cuts. Even a relatively modest change, like doubling the cap for married couples, would cost $230 billion over a decade, according to the Committee for a Responsible Federal Budget. More generous alterations along the lines of what New York Republicans have demanded could surpass $1 trillion.