WASHINGTON — Food stamp recipients are ripping off the government for millions of dollars by illegally selling their benefit cards for cash — sometimes even in the open, on eBay or Craigslist — and then asking the government for replacement cards.

The Agriculture Department wants to curb the practice by giving states more power to investigate people who repeatedly claim to lose their benefit cards.

It is proposing new rules Thursday that would allow states to demand formal explanations from people who seek replacement cards more than three times a year. Those who don’t comply can be denied further cards.

“Up to this point, the state’s hands have been tied unless they absolutely suspected fraudulent activity,” said Kevin Concannon, the department’s undersecretary for food, nutrition and consumer services.

Overall, food stamp fraud costs taxpayers about $750 million a year, or 1 percent of the $75 billion program that makes up the bulk of the department’s total budget for the Supplemental Nutrition Assistance Program.

Most fraud occurs when unscrupulous retailers allow customers to turn in their benefits cards for lesser amounts of cash. But USDA officials are also concerned about people selling or trading cards in the open market, including through websites.

Last year, the department sent letters urging eBay and Craigslist to notify customers that it’s illegal to buy and sell food stamps. USDA officials followed up last month, saying they are still getting complaints that people are using the websites to illegally market food stamps.

Both eBay and Craigslist have told the government they are actively reviewing their sites for illegal activity and would take down ads offering food stamp benefits for cash. The USDA also has warned Facebook and Twitter about the practice.

South Dakota, Oklahoma, Washington, D.C., Minnesota and Washington state have the highest percentage of recipients seeking four or more replacement cards over a year. But USDA officials said that doesn’t necessarily indicate a high rate of fraud. All states are required by law to reissue lost or stolen cards to those who are eligible for benefits.

Wyoming, Idaho, New Hampshire, North Carolina and Alabama have the lowest percentage of households requesting four or more cards in a 12-month period.

In North Carolina, the state already issues warning letters to people who request four replacement cards in a year, letting them know that officials are monitoring them closely. Dean Simpson, chief of economic family services for the North Carolina Division of Social Services, said the new rules would give her state even more of a boost in curbing food stamp fraud.

“I think it would help with the trafficking and let individuals know they are being observed and watched,” said Simpson, who oversees the state’s distribution of food stamps.

More than 46 million people receive food stamps, nearly half of them children. The average monthly benefit is $132 per person.

Benefit cards work like debit cards, allowing users to swipe them for food purchases at some 231,000 stores around the country that are authorized to take part in the food stamp program. Once a card is reported lost or stolen, it can be disabled immediately. But the USDA does not require photo identification, since several members of a family, including children, may use the cards at different times.

Concannon stressed that the USDA wants to be sensitive to vulnerable people who may lose their cards for innocent reasons. While it may sound suspicious for someone to lose a card two or three times a year, food stamp recipients include many people who are homeless or have dementia or mental illness, he said.

“Our concern is that in many instances, it may point to a trafficking issue,” he said.

Last year, about 850,000 people were investigated for possible food stamp fraud. About 2,000 stores were sanctioned for illegal conduct, and 1,200 stores were permanently removed from the food stamp program.

Large supermarkets are seldom involved in illegal activity, Concannon said. The vast majority of fraud is found in smaller shops and convenience stores.

The USDA is currently developing tougher sanctions and penalties for retailers engaging in food stamp fraud. It is also taking steps to make sure that people disqualified from the program for illegal activity are not able to use it again in other states.