The actor was told to leave with Boo and Pistol by Saturday but he is believed to be leaving today instead
The Hollywood superstar, 51, became embroiled in an customs furore when it emerged he entered the country by private jet and failed to declare to customs officers his beloved Yorkshire terriers were on board.
Minister Barnaby Joyce said the animals may yet be put down, with uncertainty surrounding whether US biosecurity officers will permit them to return to the country.
Mr Joyce said he was "seriously worried" they might not have the right permits to get back into the US and could be rendered "stateless".
"The question is if he breached our laws, then did he follow the correct laws in the US?" he said. "My worry is will the US let them back in? If not ... will they have anywhere to go?"
The reports of the couple's departure come as Liane Kent, the Gold Coast dog groomer who unwittingly alerted authorities to the tiny quarantine fugitives with a social media post, took to Facebook to defend herself against the move, saying she had the permission of Depp's minders to post the photo.
Ms Kent said that she has been subject to online abuse since she was outed as the person who inadvertently sparked the quarantine investigation.
Depp, who is currently in Australia to film the latest instalment of Pirates of the Caribbean, and his wife Amber Heard have been accused by the Department of Agriculture of breaking import laws by not declaring Yorkshire Terriers Boo and Pistol to customs when they flew in by private jet from California last month.