The Red Sun bank says it has not yet determined the maximum amount that it could charge its customers who are unable to afford to pay their monthly bills.
The bank has been offering the RedSun savings account, the new name of the Bank of New South Wales, for about a year.
It has a limit of $2,500, a deposit limit of up to $1,000 and a maximum of $500 a month.
Its biggest offer, $500 in a month, has been limited to banks and small businesses, but other customers can take it.
A deposit of $5,000 is offered for people with no more than $5500 in assets, and $3,500 for people who have more than a $3.5 million in assets.
RedSun is one of many banks offering savings accounts.
The Reserve Bank of Australia says its customers can open savings accounts of up $500 or $1 million a month for up to six months, up to two years or $2 million.
But the Bank says customers must still be able to afford their monthly mortgage repayments and their annual rent and property payments.
Read more: How much can a bank charge me and how to close an account?
If you are struggling to make ends meet and want to open a savings account of up a million dollars, the bank has a number of other ways you can open an account.
You can: open an online account at a bank branch