“We support the bill as a whole,” Ms. Coven said. “We strongly support House passage of the Senate bill.” She declined to comment on criticism of the food stamp changes.

Lawmakers from both parties and child nutrition advocates have praised many provisions of the Senate bill. It gives the secretary of agriculture the authority to establish nutrition standards for foods sold in schools during the school day, including vending machine items. And it would require schools to serve more fruits and vegetables, whole grains and low-fat dairy products.

Also, for the first time in over three decades, it would increase federal reimbursement for school lunches beyond inflation  to allow for the cost of higher-quality meals. It would also allow more than 100,000 children on Medicaid to qualify for free school meals, without filing applications.

Representative Jim McGovern, Democrat of Massachusetts, said: “I want to pass a child nutrition bill. I am committed to the first lady’s campaign. I want to be helpful. But I won’t vote for a bill that robs Peter to pay Paul. The White House needs to work with us to find a better way to offset the cost.”

Representative Barbara Lee, Democrat of California and chairwoman of the Congressional Black Caucus, said, “While I want a strong and robust reauthorization of the Child Nutrition Act, we cannot do it on the backs of the unemployed, underemployed and chronically poor.”

Mayor Michael R. Bloomberg of New York City weighed in too. He wrote that he was concerned about the health and welfare of the more than one million students in the city’s public schools. But he said that Congress should not try to help them at the expense of food stamp recipients.