In the so-called 'vishing' scam, the 16 victims were told there had been suspicious activity on their account and were advised to transfer their money to a different account.
One 53-year-old woman, who asked to remain anonymous, said she considered herself to be competent and security aware in dealing with her financial affairs but still fell victim to the "sophisticated and elaborate" scam and almost lost close to £165,000.
Police in Edinburgh and Royal Bank of Scotland (RBS) have issued a warning to people about the gang, who call up the public claiming to be from their bank.
As part of an awareness campaign all branches of the main banks will display posters reminding the public never to give out their details if they are cold-called. Police will also distribute crime prevention leaflets across the capital giving information on how to avoid becoming a victim.
Detective Inspector Arron Clinkscales said: "If you receive a call like this, do not comply. Hang up and ensure the line has been cleared before contacting police."
The woman from Edinburgh, who spoke of the fraud yesterday, said she transferred £163,499 on instruction from the criminals. Her bank managed to stop £100,000 before it hit the suspect's account, but £63,499 did go through. Police and the banks recovered £19,999, and the rest of the money, £43,500, was reimbursed by her bank.
She said: "The most important element of this fraud, which hooked us into it from the start, and subsequently kept us motivated to participate, was that, as far as we were concerned, we had called the telephone number on the back of our bank debit cards, which the fraudsters had asked us to do for our own 'security' and so that we could be 'confident' we had called the bank's fraud department.
"We did not know at the time that, when they called us, they had simply kept the line open and we were, therefore, still speaking to them and not to the bank."
She transferred money from three separate banks into what turned out to be fraudulent accounts used for money laundering in the space of two to three hours.
Chris Wilson, RBS Scotland managing director, said: "Fraudsters work by creating fear a customer's savings may be under threat.
"No bank will ever ask a customer to transfer their savings or part of their savings to another account or another bank in order to 'protect the funds'."