WASHINGTON — Republicans controlling the House are eying big cuts to food stamps as they piece together legislation to trim $261 billion from the federal budget over the next decade, hoping to forestall major Pentagon cutbacks.

The cuts to food stamps would reduce the monthly benefit for a family of four by almost $60, repealing increases that were enacted three years ago as part of President Obama’s economic stimulus. The changes would also force up to 3 million people out of the program by tightening eligibility rules, the administration estimates.

The food stamp cuts would total $8 billion over the coming year and $34 billion over a decade. The program has been expanded greatly over the past few years – enrollment tops 46 million nationwide, up from about 33 million in 2009 – and now costs about $80 billion a year.

The average monthly benefit for a family of four is about $500, according to the Center on Budget and Policy Priorities, a liberal research and advocacy group.

Food stamps are just one of the cuts Republicans want to muscle through the House as a follow-up to the tightfisted GOP budget plan approved last month. That measure is nonbinding but instructed six House committees to come up with spending reductions as an alternative to across-the-board cuts scheduled to slam both the Pentagon and domestic agencies in January. Those required cuts are a consequence of the failure of a budget “supercommittee” to agree on a deficit-reduction plan last year.

House panels are producing legislation this week as the first step in implementing the GOP’s budget plan, starting with Judiciary panel action Tuesday on a proposal that would sharply limit damage awards in medical malpractice lawsuits.