Jonathan Butcher, a senior policy analyst at the conservative Heritage Foundation, said that the rule would ensure that the government was “providing resources to the children who are in need and not providing resources to those who are not in need.”

Right now, households that receive benefits or services from another federal welfare program, Temporary Assistance for Needy Families, are automatically eligible for food stamps in 39 states, the District of Columbia, Guam and the Virgin Islands. In some of those states, households with gross incomes up to 200 percent of the poverty line — which would be about $50,000 for a family of four — are automatically eligible for food stamps. Children in those households are automatically eligible for free school meals, too; over 8 million children were directly certified for free school meals in this way in the 2016-17 school year. Not all of those who automatically qualify for food stamps or free school meals necessarily receive them though.

Under the proposal, fewer families would automatically qualify for food stamps, and in turn, fewer children would get free school meals. Children in households with gross incomes between 185 percent and 200 percent of the poverty line would no longer be automatically eligible for any food assistance at school. And children in households with gross incomes between 130 percent and 185 percent of the poverty line would be eligible for only reduced-price meals. Families would be charged a maximum of 30 cents for breakfast and 40 cents for lunch.

“Even that reduced price fee is very, very burdensome on families that are struggling to make ends meet,” Ms. Davis said.

The Agriculture Department estimates that schools serve about 34 million free and reduced-priced breakfast and lunch meals a day. Critics of the change worry that losing free school meals would negatively affect the children’s academic performance and health.

Healthy school meals “are proven to support academic success, obesity prevention and overall student health,” said Gay Anderson, the president of the School Nutrition Association.

Critics also worry that the proposal would worsen the debts already being accrued by families who have a hard time paying their school lunch fees. Episodes of “lunch shaming” — when children are publicly reprimanded for being unable to pay a school lunch bill, including by having their food thrown away — have occurred and have received news media coverage. This month, a Pennsylvania school district sent 1,000 letters to parents with outstanding lunch debt, threatening to take them to court and warning that their children could end up in foster care.

A cafeteria worker in New Hampshire said she was fired this year after she said she fed a student who could not pay. Other school districts have been criticized for serving students alternative meals, like cold cheese sandwiches, or for not feeding students at all if they are behind in their lunch payments.