A spokeswoman for BNZ said "maintaining our customers' privacy and meeting our obligations under the Privacy Act is very important to [us]."
"We have a very strict set of protocols when it comes to requests to release information and at this stage we are not looking to change that process. An official written police request specifying the act and grounds is required for all requests and each is then considered on a case-by-case basis," she said.
An ANZ spokesman said it would "take privacy very seriously" and comply with the Privacy Act
The spokesman said that unless police obtained information by using a production order, search warrant or court order, they were guided by the Code of Banking Practice and Letter of Agreement between the New Zealand Bankers' Association and the New Zealand Police when deciding whether or not to release customer data.
That code lists the laws under which banks may be required to release information.
Kiwibank would not comment, referring instead to the New Zealand Bankers' Association.
The New Zealand Bankers' Association said that in handling information requests, "the Police must provide the bank with the legal basis on which the information is requested".
Westpac claimed it had the power to release information when customers signed the bank's "terms and conditions" - a move it claimed voided customers' rights.
It was an argument unsuccessfully made by the bank in its attempt to defend itself over the disclosure of Hager's details to police investigating his Dirty Politics book.
The ruling stems from a police inquiry into the identity of Rawshark, the hacker who provided Hager with information used in the 2014 book. The book argued that John Key's Government was using right-wing bloggers as attack dogs to go after opponents.