The bank said for most of those impacted members the underpaid amounts were expected to be $50 or less.
But exact numbers and amounts would not be known until the correction had been processed by the Inland Revenue.
"ANZ is making a claim on behalf of impacted KiwiSaver members to Inland Revenue for the additional member tax credits to be credited to those customers' KiwiSaver accounts.
"In addition, ANZ will credit their KiwiSaver accounts with the investment returns that would put them in at least the same position they would have been if the error had not occurred."
The bank said it expected payments to be completed by August and affected members would receive a letter about it at that time.
An ANZ spokesman said it discovered the errors during a recent in-depth check of its systems and processes.
"We were relying on internal systems and processes as they were originally set up. However, the issue was discovered during a recent in-depth check of our systems and processes and we acted quickly to put it right."
The problems relate to three errors which occurred between 2009 and 2016;
• The start date of some KiwiSaver members was set as their actual joining date, instead of the first day of that month. This meant their pro-rata member tax credit amount was slightly underestimated for that year. All subsequent years were correct.
• For a number of other members who switched to ANZ from other providers, their member tax credit was based on the time they were in an ANZ scheme only. All subsequent years were correct.
• A one-off processing error in 2009 involved a small number of members who made a voluntary contribution by direct debit to their KiwiSaver account. The money went into their account, but that contribution was not included in the calculation of the member tax credit for that year.
The spokesman said there had been changes to its processes which meant it would not happen again.
ANZ manages more than $10 billion of KiwiSaver investments on behalf of 700,000 people.
There are 2.7 million people signed up to KiwiSaver with around $40 billion invested.