AT first, the constant attention seemed sweet and very special.

Janet N. Cook, a church secretary in the Tidewater, Va., area, had been a widow for a decade when she joined an Internet dating site and was quickly overcome by a rush of emails, phone calls and plans for a face-to-face visit.

“I’m not stupid, but I was totally naïve,” said Ms. Cook, now 76, who was swept off her feet starting in July 2011 by attention from a man who called himself Kelvin Wells and described himself as a middle-aged German businessman looking for someone “confident” and “outspoken” to travel with him to places like Italy, his “dream destination.”

But very soon he began describing various troubles, including being hospitalized in Ghana, where he had gone on business, and asking Ms. Cook to bail him out — again and again. In all, she sent him nearly $300,000, as he apparently followed a well-honed script that online criminals use to bilk members of dating sites out of tens of millions of dollars a year.

Many of those targeted are women, especially women in their 50s and 60s, often retired and living alone, who say that the email and phone wooing forms a bond that may not be physical but that is intense and enveloping. How many people are snared by Internet romance fraud is unclear, but between July 1 and Dec. 31, 2014, nearly 6,000 people registered complaints of such confidence fraud with losses of $82.3 million, according to the federal Internet Crime Complaint Center.