The median loss for romance fraud victims in their 20s was $770. People in their 50s reported losing twice as much. The losses reached $3,000 for victims in their 60s and $6,450 for those in their 70s.

“We’ve heard of people refinancing their homes and cashing out retirement accounts,” Ms. Nofziger said. “Scammers go where the money is, and criminals know that older adults have the majority of assets in the United States.”

Last year, federal prosecutors brought a number of alarming romance cases. A 76-year-old widow in Rhode Island transferred more than $660,000 to bank accounts she thought belonged to a U.S. Army general in Afghanistan. (Posing as a military member is another red flag, along with overseas locations.)

In Oklahoma, 10 Nigerian and United States citizens were indicted in a fraud ring targeting multiple victims in three states. A grand jury in Georgia indicted a man accused of bilking a Virginia woman, who had a large trust, of $6.5 million.

Ms. Floren may qualify as one of the luckier victims. As “James Gibson” was leaving for Europe, he suddenly called, saying his Netflix card had expired. “He really wanted to be able to watch movies on the plane,” she recalled. “Would I please go to a Walmart and buy him a $100 Netflix card?”

Gift cards, untraceable and available everywhere, have become the currency of choice for scammers, Ms. Nofziger said. But they may also ask victims to open a bank account and provide access to it, or to ship iPhones.

Ms. Floren bought a gift card, reading her apparent suitor the number. Three days later, he called again, claiming to have left a bag of expensive tools in a cab. “He was hysterical on the phone,” she said. The tools were worth $4,000, but he’d found replacements for just $2,600. Would she send him the money?