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 Boxing Day.
"He was pretty emaciated so I crushed up a dry biscuit and left it out there for him," he told the AP news agency.
