If the U.S. Department of Agriculture approves, Minnesota would be the first state to impose such restrictions.
The change would still need the approval of the Legislature, where some anti-poverty activists call it a mean-spirited intrusion into the cupboards of the poor.
In a letter sent Monday, the state Department of Human Services described the ban as part of a broader state effort to improve eating habits.
"It is inconsistent to encourage healthy nutrition and simultaneously allow the purchase of candy and soft drinks with food stamps," assistant commissioner Maria Gomez wrote.
Lawrence Rudman, spokesman for the USDA regional office in Chicago, said that the agency had not seen the letter as of Friday and that officials will have 60 days to respond once they do.
The federal government already restricts the use of food stamps somewhat. Recipients cannot use food stamps to buy liquor or tobacco, or hot foods such as rotisserie chicken.
Republican Gov. Tim Pawlenty first proposed the junk-food ban more than a year ago.
As for the tricky task of deciding which foods are healthy and which aren't, Minnesota would simply apply the definitions already in the state's tax code. Minnesota taxes candy and soda but exempts most other store-bought foods.
However, the tax code is full of inconsistencies. The state, for example, taxes Hershey's bars but not Kit Kat bars, because anything made with flour is not considered a candy. It taxes gum but not licorice. It taxes marshmallows but not ice cream bars. Also, potato chips would not be banned.
Colleen Moriarty, executive director of Hunger Solutions, which represents Minnesota food banks, said it would be better to simply try to educate food stamp recipients about nutrition.
"I think it's a bad idea to regulate people instead of empowering them," she said.