THANE, India—Minaz Husain Ladaf couldn’t believe his luck this summer when he was offered a 40% bump in salary and other perks to work for what he thought was the U.S. government. But when he went to his new workplace to pick up his paycheck this week, he was picked up by the police instead.

Mr. Ladaf and dozens of fellow call-center workers have been locked in dusty jail cells in this booming Mumbai suburb since Wednesday. Police say they managed a telephone “scam center” where close to 700 workers called targeted Americans,...