The victims are customers of the Bank of Scotland
Bank customers have been warned not to give out their personal information over the telephone after an elaborate scam was uncovered.
Police believe a gang of criminals has managed to divert calls made by some customers to the Bank of Scotland.
The call is then intercepted by the scammers, who take the customer's details and use the information to steal money from their account.
Up to 25 people across Scotland are thought to have been hit.
Among them was a 66-year-old woman in Perth who had £1,500 stolen last month, while a 78-year-old woman was contacted and duped into giving personal details and disclosing her International Bank Account Number, the international standard for identifying bank accounts across borders.
A short time later the fraudster attempted to transfer £27,000 out of her account into a bank account in India, but the Bank of Scotland intervened and the transaction was not completed.
Both women had been contacted by a man with an Asian sounding accent, who claimed to be from Bank of Scotland. The man had the women's names and knew they had accounts with the bank.
Tayside Police said it appeared the scammers had somehow managed to tamper with their victim's phone lines, allowing them to divert the call when the women rang back the genuine Bank of Scotland telephone number.
One theory being investigated is that a former call centre employee is involved in the scam.
Det Ch Insp Bruce Kerr said: "We are working closely with the bank in a bid to track down those responsible.
''Those involved in this fraud would appear to be well organised as, on both occasions, the man calling already had some personal information about the account holders, helping to lull them into a false sense of security.
"The man claims that the account holder has been overcharged on their respective accounts and that he needs to confirm account details before he can proceed.
"Sadly, some people have given out or confirmed account details not realising that they are not actually talking to someone from the bank."
He urged people to never disclose or confirm any personal information over the telephone, particularly bank details.
Anyone with an doubts or concerns should go into their local branch in person, he added.
A Bank of Scotland spokesman said: "As you would expect, Bank of Scotland has a number of systems in place to detect fraudsters. We work hand in hand with the police on this.
"We would firmly echo the advice of the police that customers should never disclose or confirm any personal banking details over the phone.
"Customers should be very wary of anyone calling them, asking for personal details. Your bank will never call you and ask you to give out account details to them."