WASHINGTON—A long-running political battle over how much to cut food stamps could take a new turn this week, as the House is expected to weigh a Republican measure to scale back the program further than Democrats say they can accept.

The House GOP bill would cut spending on the Supplemental Nutrition Assistance Program by about $40 billion over 10 years, according to congressional aides from both parties. That is twice the amount that House Republicans aimed to cut as part of a bill scuttled in June amid moves by House conservatives...