WASHINGTON — Cuts to the nation’s food stamp program enacted this year are only affecting four states, far from the sweeping overhaul that Republicans had pushed, an Associated Press review has found.

As a result, it’s unclear whether the law will realize the estimated $8.6 billion in savings over 10 years that the GOP had advertised.

A farm bill signed by President Barack Obama in February attempted to save money by scaling back what lawmakers called a loophole in the food stamp program that entitles low-income families to more food aid if they participate in a federal heating assistance program. States were giving some people as little as $1 a year in heating assistance so they could get more food aid. It’s called “heat and eat.”

Among the 16 states that allow the practice or some form of it, 12 governors have taken steps to avoid the food stamp cuts.

“Government’s role is to help people help themselves, and these steps are necessary to help our most vulnerable residents and families meet their most basic needs,” Massachusetts Gov. Deval Patrick said when he announced his state’s move earlier this year.

The farm bill was held up for more than two years as conservatives insisted on cutting the nation’s food stamp program, which now serves 1 in 7 Americans at a cost of around $80 billion a year. The roughly 1 percent cut was a compromise between Republicans who had hoped for far larger cuts and Democrats who didn’t want to see any cuts at all.

The states’ workaround — mostly by Democratic governors — has infuriated Republicans who pushed the cuts. In March, House Speaker John Boehner, R-Ohio, called the states’ moves “fraud.” House Agriculture Chairman Frank Lucas, R-Okla., and House Energy and Commerce Chairman Fred Upton, R-Mich., have asked the Obama administration to “hold states accountable” for dodging the cuts.

The governors say they are following the law while preserving crucial benefits for their neediest citizens.

The new law says that people can’t get the higher food benefits unless they receive more than $20 a year in heating assistance, which lawmakers hoped would be too expensive for states to pay. But the governors in 12 states and the mayor of the District of Columbia have said they will find a way. Most will use federal heating assistance dollars. At least one state, California, will use its own money.

As of now, the cuts will only affect Michigan, Wisconsin, New Jersey and New Hampshire. All but New Hampshire have Republican governors.

There are about 1.8 million households that receive food stamps in those four states, out of almost 23 million households nationwide.

It’s unclear how many people will be affected. Officials in Wisconsin, New Jersey and New Hampshire said they don’t track that number. Michigan officials say around 20 percent of the state’s recipients, or around 170,000 households, participated in the “heat and eat” program and will see cuts.

Bob Wheaton, spokesman for the Michigan Department of Human Services, says the state didn’t want to “create a new loophole even beyond the loophole that previously existed” and draw down federal heating benefits for others in the cold-weather state. He said the average decrease will be around $76 a month for a family of four, starting in November. That amount varies by state.

Terry Smith, director of New Hampshire’s family assistance programs, said his state’s decision “was not to deplete an already tenuous LIHEAP allocation in our state and take needed heat from people.”

LIHEAP is the Low Income Home Energy Assistance Program, and it is paid to states as federal grants each year. New Hampshire did not give recipients $1 payments but did allow a LIHEAP application to qualify them for higher food benefits. The farm bill’s change in policy will discontinue that practice.

The states that are using that federal heating assistance money to avoid the food stamp cuts say they believe they can do it without significantly reducing heating aid to others who need it, even without more money from the federal government. Peter Merrill, the deputy director of MaineHousing, says he estimates that maintaining the food stamp benefits will only reduce federal heating assistance payments to Maine residents by about $4 a year on average.

In Washington state, residents will see food stamp benefits reduced briefly, in November and December, due to a backlog in getting their computer systems running. A spokeswoman for the governor said the state will reinstate the higher heating assistance payments in January, once the backlog clears, and 200,000 households will see their benefits go back up.

On Capitol Hill, Republicans say the states’ decisions don’t mean the farm bill cuts are obliterated. A GOP memo from the House Agriculture Committee staff notes that some states may reverse their decisions to avoid the cuts, especially as current recipients move off the rolls. And the Congressional Budget Office, which figures out how much bills cost, accounted for some states bowing out when coming up with its $8.6 billion estimate over 10 years. But the CBO hasn’t said whether it accounted for high-population states like California, New York and Pennsylvania maintaining the higher food stamp benefits.

Other states that have dodged the cuts are Connecticut, Delaware, Montana, Oregon, Rhode Island and Vermont.

Pat Baker of the Massachusetts Law Reform Institute, an advocacy group that focuses on poverty issues, says the “heat and eat” recipients are often elderly or disabled, sometimes living in apartments where utilities are included but the rent is higher. “This would be a significant loss in nutrition benefits to the lowest-income and neediest residents,” she says.