"It's time that Pistol and Boo buggered off back to the United States," added the minister. "He can send them back or we will have to euthanase them."
Asked if his tough stance may affect Depp's view of him, Joyce replied: "I don't think Mr Depp will be inviting me to the grand opening of the Pirates of the Caribbean."
Depp, who stars as Captain Jack Sparrow in the Disney franchise has been shooting on the Gold Coast tourist strip in Australia's east.
"Pirates of the Caribbean: Dead Men Tell No Tales" is the fifth instalment in the blockbuster series, which has raked in more than US$3.6 billion at the box office.
Sydney (AFP) - Johnny Depp's pet dogs Pistol and Boo face being put down unless they "bugger off" back to Hollywood after he brought them into Australia illegally on his private jet.
The actor, who is in the country to film "Pirates of the Caribbean: Dead Men Tell No Tales", was given the grim warning by Agriculture Minister Barnaby Joyce on Thursday.
"Just because he's Johnny Depp does not mean he's exempt from Australian law," Joyce told reporters.
Australia has strict animal quarantine laws to keep disease at bay, which Depp appears to have ignored, arriving in Brisbane last month with the Yorkshire terriers in tow without declaring them to customs.
"There is a process if you want to bring animals -- you get the permits, they go into quarantine and then you can have them," said Joyce, with customs authorities expected to seize the dogs imminently.
He said he became aware that Depp had the dogs with him after they were spotted being taken to a groomer.