He has claimed to people that their accounts are outstanding by a few pounds and he requires their bank details to settle the amount.
The States of Jersey Police are working with Jersey Telecom to investigate the crime and try to identify where the calls are coming from.
At this stage it is believed they are coming from a New Zealand number, but this may not be genuine.
After engaging the caller in conversation, the bogus caller then asks the person for their bank details.
Some of the calls made have been quite lengthy.
Jersey Telecom reported the scam to the police yesterday and are now reassuring customers that they are taking steps to find out who the bogus caller is.