The best hope for a deal to avoid the “fiscal cliff” may lie with the alternative minimum tax, an obscure provision of the tax code that is about to become alarmingly relevant to millions of middle-class taxpayers.

Unless Congress acts by the end of the year, more than 26 million households will for the first time face the AMT, which threatens to tack $3,700, on average, onto taxpayers’ bills for the current tax year. Because those people have never paid the AMT, they have no idea they are in its crosshairs — put there by a broader stalemate over tax policy that has kept Congress from limiting the AMT’s reach.

Forget about the much-publicized tax hikes set to take effect for 2013 — if you have a couple of children and annual income over $75,000, chances are good that your taxes are on track to go up substantially for 2012.

Residents of high-cost urban areas, including Washington, would be hit hardest, with about 2 million households in Maryland, Virginia and the District in line to face the AMT for the first time, by official estimates.

Unlike most tax increases in the fiscal cliff, including the expiration of the George W. Bush-era income tax cuts, the AMT bill would come due almost immediately. And tax experts say it would be extremely disruptive to try to fix the AMT after the 2012 tax year closes Dec. 31.

Who pays the Alternative Minimum Tax

Officials with the Internal Revenue Service and the Treasury Department declined to comment on the impact of adjusting the AMT after December. But congressional tax aides said the IRS has advised Congress that trying to fix the AMT after the filing season begins in January would lead to processing delays of more than two months for nearly half of all returns — significantly postponing the delivery of refunds.

“That would be a disaster, an unmitigated disaster for the taxpayers of the United States. It’s just not possible to do that,” said Nina Olson, national taxpayer advocate at the IRS. Olson noted that many people count on their ­refunds, which average around $3,000, to cover immediate needs. For example, she said, many utilities do not do shut-offs for nonpayment in January and February, because they know people will use their tax refunds to get caught up on their heating bills.

Lobbyists and aides in both parties say it is hard to imagine Congress letting the new year arrive without legislation to restrict the AMT. Optimists say the need to fix the problem could force Republicans and Democrats to come together on a plan to address the fiscal cliff, which comprises $500 billion in tax hikes and automatic spending cuts set to take effect in January, potentially snuffing out the country’s economic recovery.

The alternative minimum tax was created in 1969 to prevent the super-rich from using credits, deductions and other shelters to avoid taxes altogether. In simple terms, it’s a flat tax with two brackets, 26 percent and 28 percent. Breaks for dependents, medical expenses, and state and local taxes are all disallowed. Instead, taxpayers get a single big deduction, called the AMT exemption, which usually rises with inflation. Taxpayers who owe more under AMT rules than under normal tax rules must pay the higher amount.

Over the decades, the AMT has ceased to affect the extremely rich, because their tax bills are higher than the AMT rates. Instead, the inflation-adjusted AMT has come to target 4 million to 5 million taxpayers with annual incomes between $200,000 and $1 million.

This year, however, a gridlocked Congress has failed to approve the inflation “patch” that prevents millions of people from falling into the AMT’s grasp. The last patch expired in December. If a new one is not enacted, the AMT will hit 31 million taxpayers this year, reaching deeply into the middle class.

In the District and Maryland, where about 6 percent of taxpayers have become accustomed to paying the AMT, the figure would jump to 41 percent in D.C. and 38 percent in Maryland, according to official estimates. In Virginia, where 4 percent of taxpayers have routinely paid the AMT, the figure would increase to 28 percent.

Residents in other high-cost areas would also get socked. In New Jersey, for instance, where people are still digging out from Hurricane Sandy, taxpayers would face a fresh burst of bad news when they break out the TurboTax early next year. More than half of all Garden State households would owe unexpectedly large tax bills, the highest percentage in the country.

Nationwide, nearly one in five taxpayers is in line to pay the AMT. And while the levy is causing bipartisan nightmares, urban states — which tend to lean Democratic — are by far the most vulnerable.

That poses a challenge for Democrats planning to use the fiscal cliff as leverage to force Republicans to raise taxes on the wealthy to help reduce the federal budget deficit. President Obama has threatened to veto legislation that extends the Bush tax cuts for the wealthiest 3 percent of households, a top GOP priority.

If Obama wins reelection Tuesday and Republicans refuse to give in, Democrats say they are prepared to sail over the cliff and let the Bush tax cuts expire for everyone for the 2013 tax year. That would put them in position to press legislation in January to restore the Bush tax cuts — probably in a different form — for taxpayers earning less than $250,000 a year.

But the AMT could throw a wrench into those plans, because there is no easy way to deal with the levy once the cliff is breached.

For the moment, party leaders are blaming each other for not addressing the problem. To Republicans, the AMT is another reason to “extend all the expiring tax policy so we can overhaul the tax code to create jobs, spur economic growth and get rid of the AMT once and for all,” said Antonia Ferrier, spokeswoman for Sen. Orrin G. Hatch (Utah), the senior Republican on the Senate Finance Committee.

Democrats, meanwhile, accuse Republicans of holding AMT payers hostage in the pursuit of tax advantages for millionaires. “The fact that Republicans are willing to risk not doing an AMT patch — which has always been bipartisan and relatively noncontroversial — shows to what extreme they’re willing to go to protect the wealthy few,” said Sen. Charles E. Schumer (D-N.Y.).

According to estimates by the Tax Policy Center, more than half of all married couples will owe an additional tax of around $4,000 unless Congress acts. And more than a third of families with children will fall subject to the AMT, with parents of three or more children facing an extra tax of $4,700.

Among married couples with at least two children and adjusted gross income between $75,000 and $100,000, the center estimates that 84 percent will face a significantly higher tax bill this year because of the AMT.

There is also the possibility that some people could face penalties for failing to withhold a sufficient amount throughout the year to cover a tax they did not know they would owe, said Leonard Burman, a tax expert at Syracuse University.

“But politically, it’s inconceivable that Congress would let that happen,” Burman said. “They’re all playing this dangerous game of chicken.”