Employees of the St. George Bank in Caringbah found the snakes in the kitchen area at the back of the building, The Daily Telegraph of Sydney reported. The area is not open to customers.
The Australian brown is one of the world's most venomous snakes, capable of causing death almost instantaneously if it bites.
Lara Daniels, a spokeswoman, said the branch closed down as soon as the snakes were spotted.
Caringbah is about 20 miles south of Sydney in New South Wales.