BNZ borrowers will also need to have at least 20 per cent equity and be borrowing at least $400,000 or more.
But to get the full $20,000 a borrower will have take out a loan for at least $2 million. There aren't likely to be many of those.
The moves come as the value of new lending plummets compared with a year earlier.
Last week Reserve Bank figures show $6.8 billion was lent out by the banks in May, down from $8.9b in May 2021 - although it was still up on the $4.3b done in May 2020 when the country was just coming out of its first nationwide Covid-19 lockdown.
Mortgage rates have risen steeply in recent months and at least three major banks have put a pause on lending to low deposit borrowers.
Karna Luke, BNZ executive customer, products and services, said the bank hoped to give people more cash in hand to cover the costs of buying and selling a property.
For a customer buying around the average property price in Auckland he said this would mean over $10,000 in cashback.
Latest Barfoot and Thompson figures put the average price of an Auckland house sale at $1,158,464 in June down 2.6 per cent on May.
While the average first home buyer nationwide, borrowing around $530,000 for their first home, could get $5300, as long as they had at least a 20 per cent deposit.
The offer ends on August 5 with the mortgage needed to be drawn down by August 26 to be eligible.