The proposed rule by the Agriculture Department, coming on the former first lady’s birthday, would give schools more latitude to decide how much fruit to offer during breakfast and what types of vegetables to include in meals. It would also broaden what counts as a snack.

A spokeswoman for the department said that it had not intended to roll out the proposed rule on Mrs. Obama’s birthday, although some Democratic aides on Capitol Hill had their doubts. Food companies applauded the proposal, while nutritionists condemned it, predicting that starchy foods like potatoes would replace green vegetables and that fattening foods like hamburgers would be served daily as “snacks.”

“Schools and school districts continue to tell us that there is still too much food waste and that more common-sense flexibility is needed to provide students nutritious and appetizing meals,” Sonny Perdue, the agriculture secretary, said in a statement. “We listened and now we’re getting to work.”