The elaborate swindle is being investigated by the London-based National Fraud Intelligence Bureau (NFIB).
A con man who advertised a city centre flat for rent on the Gumtree website convinced his victim to transfer cash to one of her friends.
The fraudster then asked for a copy of the receipt to be sent to what is thought to be a fake email account.
The victim used Western Union to move £765 to her friend.
However, the cash was collected from a Western Union branch 400 miles away in Harrow, London before it could be claimed in Glasgow.
The con man spent weeks emailing the young Greek woman with a detailed back story in which he claimed to be working overseas.
After sending pictures of a plush flat he claimed was in Ingram Street, he insisted she provided evidence of her funds in order to arrange a viewing.
The University of Strathclyde graduate, who asked to remain anonymous, said: "I went to Western Union to make sure that the only person who could pick up the money was the person it was meant to go to.
"They reassured me and I transferred the money on the Thursday to my friend, then sent the receipt in an email to the guy with the flat.
"He replied and said he'd see me on Sunday at the flat.
"The money was collected on the Friday in London and I have never heard from the guy since."
The victim spoke to police and sought the support of Glasgow lawyer, Andy Bradley, who has taken on the case.
He said: "This will be a concern for customers of a corporation handling such large sums of money.
"It is unclear whether such an elaborate crime is a one off or has happened repeatedly.
"Customers will wish reassured that the company is doing whatever it can to ensure the utmost security, both online, and at the point of handover."
A spokeswoman for Police Scotland said: "Unfortunately some criminals will try to exploit visitors by advertising bogus properties for rent."
The victim also complained to Gumtree and the website's customer service team apologised and reported the scam to the National Fraud Intelligence Bureau.
A spokesman for Action Fraud, an arm of City of London Police, said: "The report is with the NFIB for assessment and to establish if the details in the report can be linked with other reports of crime, or to an identifiable suspect."
A spokeswoman for Western Union said: "We are investigating into this issue."
The Evening Times tried several times to contact the fraudster's e-mail account but he did not respond.