As more details from Johnny Depp's legal wrangling with his former money managers emerge, Australian Deputy Prime Minister Barnaby has hinted that the Hollywood actor may have another court case on his hands over his failure to declare his dogs – Pistol and Boo – to customs in 2015.

According to the politician, who had said the presence of the canines defied the country's tough quarantine laws and that they should "bugger off back to the States", an investigation may be launched to determine whether the 54-year-old actor knowingly smuggling the Yorkshire terriers into the country.

"If the allegation is correct, there's a word for that - it is called perjury," he told the Australian Broadcasting Corp.

"We're an island continent and we take biosecurity very seriously and it doesn't matter if you think that you're Mr Who's Who of Hollywood, you're going to obey our laws." He also referred to Depp as a "dipstick".

The controversial case has been inadvertently reignited by TMG. The company, embroiled in a legal dispute with Depp over the alleged mismanagement of his fortune, stated in legal filings that he was "fully aware that he was illegally bringing his dogs to Australia" when he took them Down Under in 2015.

According to the documents, Depp "falsely claimed to authorities and in public press interviews that the incident was a big misunderstanding because he supposedly believed his staff had obtained the necessary paperwork".

However, legal experts have ruled out a perjury charge, telling Guardian that because Depp made no representations in Australia, he could not be prosecuted for "attempts to obstruct, to prevent, to pervert or to defeat the course of justice in relation to a judicial power".

The dog saga ended with his then partner Heard pleading guilty to falsifying immigration documents and the former couple making a bizarre video apology. They were also placed on a A$1000 (£600, $760) one-month good behaviour bond.

"Australians are unique: both warm and direct," he said in the now infamous video after accompanying Heard to the court. "When you disrespect Australian law they will tell you."