The bird - a parrot of the Kea variety - made its move while the bus was stopped along the highway to Milford Sound on South Island, and the driver was looking through the compartment. Milford Sound, which runs inland from the Tasman Sea and is surrounded by sheer rock face, is part of Fiordland National Park, a world heritage site and major travel destination.
Police told the newspaper the passport has not been recovered and is unlikely to be located in the vast Fiordland rain forest.
"My passport is somewhere out there in Fiordland. The Kea's probably using it for fraudulent claims or something," the passport owner, who did not want to be named, told the newspaper.
A replacement passport from the British High Commission in Wellington could take six weeks and cost up to $250.
"I'll never look at a Kea in the same way," the man was quoted saying.
Kea, the world's only snow line-dwelling parrot, are widely known as inquisitive birds who appear to take delight in attacking rubber items like windshield wiper blades.
Native to New Zealand, the birds are found only in or near South Island mountains, where they live in high-altitude beech forest and open sub-alpine herb fields that stretch up into the snow line.
Covered mainly in brown and green feathers, they have large flashes of bright orange feathers under their wings.