The bird reportedly went missing during a race in the US state of Oregon in late October, before turning up in Melbourne almost two months later.
But officials say the pigeon, which has been named Joe, poses a “direct biosecurity risk” to Australia’s bird population and poultry industry.
The bird will be caught and euthanised.
Melbourne resident Kevin Celli-Bird says he found the pigeon in his back garden on 26 December.
“He was pretty emaciated so I crushed up a dry biscuit and left it out there for him,” he told the AP news agency.
Some internet research led Mr Celli-Bird to discover that the bird, which is registered to an owner in Alabama, was last seen during a pigeon race in the western US state of Oregon.
But after news of Joe’s appearance made headlines in Australia, Mr Celli-Bird was contacted by officials concerned about the threat of infection.
The pigeon has not yet been caught, but the Department of Agriculture, Water and the Environment says it will have to be put down because of the danger of infection to local birds.
“Regardless of its origin, any domesticated bird that has not met import health status and testing requirements is not permitted to remain in Australia,” a department spokesperson said in a statement.
“The only possible outcome to manage the biosecurity risk is humane destruction of the bird.”
It is not clear how the bird managed to make the 8,000-mile journey from the west coast of the US to southern Australia, but officials believe he is likely to have hitchhiked on board a cargo vessel.
While it is possible to legally bring pigeons into Australia, the process is difficult and can cost tens of thousands of dollars, and none have been legally imported from the US in over a decade.
Joe the pigeon is not the first animal to face trouble from Australia’s strict animal import laws.
Actors Johnny Depp and his then-wife Amber Heard had to issue a video apology after illegally bringing their dogs Pistol and Boo into the country on their private jet in 2015. (Courtesy BBC)