Depp appears to have taken the hint, and is thought to be arranging for the two Yorkshire Terriers, Pistol and Boo, to be whisked out of the country on a private jet.
Depp's dogs were put on notice after the actor violated Australia's famously strict biosecurity
laws by bringing them in on a private flight and without the proper documentation.
On Thursday the star, who is in Australia filming the latest installment of the "Pirates of the Caribbean"
franchise, was told that he had to get Pistol and Boo out of the country, Australia's Minister of Agriculture, Barnaby Joyce, said in a televised statement.
Joyce said that a 51-year-old man, Jonathan Christopher Depp, decided to bring two dogs to "our nation, despite not getting the proper certification and the proper permits required. Basically it looked like he snuck them in."
Joyce told Australian national broadcaster ABC on Friday that it looked like Depp was ready to comply with the order.
"I'm informed that Pistol and Boo are preparing to fly on a private jet back to the Unites States, which is the best news that I've got," he said in a radio interview
. "Obviously there's an investigation into how they came out to Australia... Mr. Depp decided he can step round our nation's laws."
Threat of death
A day earlier the minister had started a countdown for Depp, and his wife, Amber Heard, who is traveling with him, to get Pistol and Boo out of the country -- or risk the authorities putting them to sleep.
"Now Mr. Depp needs to take his dogs back to California or we're going to have to euthanize them. He's now got about 50 hours (out of a 72-hour notice period)."
The minister said that the Department needed to crack down on this high-profile case to make a point about Australia's stringent animal import laws.
"If we start letting movie stars, even if they've been 'Sexiest Man Alive' twice, to come into our nation then why don't we just break the laws for everybody. It's time that Pistol and Boo buggered off back to the United States. After that I don't expect to be invited to the opening of 'Pirates of the Caribbean.'"
After the actor allegedly smuggled the two pooches on his private jet he had an assistant take them to a dog groomer; it was here that authorities were alerted to their presence.
The grooming salon, Happy Dogz in Maudsland, Queensland, posted a grainy photo of Depp with one of the dogs on its Facebook page
, along with the message: "It's an honour to be grooming Johnny Depp and Amber Heard's two Yorkshire Terriers."
The Facebook post attracted more attention that intended, and after the Australian government's threats to kill them, the dogs' predicament quickly started trending on Twitter with the hashtag #WaronTerrier.
Others made a wry connection between the treatment of Depp's dogs and Australia's attitude towards asylum seekers.
Joyce also raised the specter of the dogs becoming "stateless" after it emerged that the dogs had not undergone the proper exit procedures when leaving the U.S.
He said that his counterparts in the U.S. could refuse to allow the animals back into the country.
"I understand that the (private jet's) manifest made no recognition that they were carrying dogs. So they obviously got on the plane in the United States without telling the United States authorities that (the dogs) were leaving and I suppose the U.S. now has to ask the questions... did you have the proper permits to take dogs out of the States? If technically you've absconded from the process... you could have stateless dogs."
The Department of Agriculture told CNN that they were not able to comment on individual cases, but a statement on the its website confirmed that it "became aware of an illegal animal importation on Tuesday 12 May."
The statement continued: "A biosecurity officer attended a Gold Coast property on 13 May and found two illegally imported dogs."
Department of Agriculture animal import laws require dogs spend at least 10 days in a government quarantine facility once arriving in the country. The strict laws are to prevent the spread of non-native diseases such as rabies, ehrlichia, leishmania, leptospirosis and internal and external parasites.