In 2015, Representative Adam Kinzinger had an unusual visitor at his constituent office inside a bus station in Rockford, Ill. A woman from India had flown to meet Mr. Kinzinger, claiming that she had developed a relationship with him on Facebook.

“She waited around in that bus station for two weeks for me to show up, and I didn’t,” he said in an interview. “She’s a poor lady, too. It took all her money to fly from India to me.”

The episode was just one of the bizarre interactions that Mr. Kinzinger said he had had over the past decade with women around the world who believed they were dating him.

That is because Mr. Kinzinger, a Republican and a lieutenant colonel in the Air National Guard, is one of what are probably thousands of United States service members who have been ensnared in a widespread fraud that has played out for years on Facebook, Instagram and other social networks and dating sites. Swindlers impersonate service members online to lure victims into false romances and then cheat the victims out of their savings.