The con artists have phoned pensioners at home claiming to be carrying out investigations on behalf of their bank.
In one incident, an 81-year-old West Lothian woman called police after she was asked to attend a branch, withdraw money and put it into another account.
A 76-year-old woman, also from West Lothian, received a call from a man purporting to be from her bank who wanted to discuss her being overdrawn. Despite pressure from the caller, she did not reveal any personal information.
A police spokesman said: “If you are uncertain as to the authenticity of a caller claiming to be from a bank, hang up and dial your bank’s customer service number.”