"Gerry Black" said she had to give him her bank account number if she wanted to get the money.
When she asked him if it was all a scam, he hung up.
"I've heard of scams, but it's never happened to me," she said. "My concern is the old people. It really breaks my heart."
New Zealand Bankers' Association chief executive Sarah Mehrtens said: "We are not aware of an organisation called the 'New Zealand Banking Association', but the name is very similar to that of the New Zealand Bankers' Association."
"If the caller appears to have your bank account or credit-card details, do not provide them with any information and contact your bank or credit-card provider immediately."
Scammers have also been calling to supposedly undertake bank-satisfaction surveys.
The New Zealand Bankers' Association released a press release warning people against the issue last week. They have received no further inquiries or leads as to who the scammers are.