Prison officers became aware of the escape while conducting a morning check of inmates at the Kerobokan penitentiary in Bali's capital, Denpasar, said Putu Ika Prabawa, an officer at Bali's Kuta Utara police station.
Last month, more than 440 prisoners escaped from an overcrowded prison on Sumatra island when they were let out of their cells to take part in Friday Muslim prayers.
Bali police were interviewing witnesses into Monday night as they continued the search for an Australian man and three other foreigners believed to have tunnelled out of Kerobokan prison.
Davidson was jailed for a year last August after he was found guilty of misusing a travel document belonging to someone else.
He had only 10 weeks still to serve of his sentence, according to Fairfax.
Police said immigration and airport authorities had been alerted and a manhunt was underway.
The notorious Kerobokan Prison has been home to many high-profile foreign criminals, including the so-called "Bali Nine" - eight men and a woman from Australia who were convicted in 2006 of smuggling heroin.
When he did not attend, an arrest warrant was issued.
The Australian fugitive had previously described the conditions in Kerobokan during his sentencing previous year.
Tony Nainggolan named the Malaysian Tee Kok King along with Australian Shaun Edward Davidson, Bulgarian Dimitar Nikolov and Indian Sayed Muhammad.
He said he had been expecting a "living hell" when he was incarcerated in April past year after being named a suspect.
But he came to the attention of authorities in March a year ago when he was staying at Rabasta Hotel in Kuta.
He had only two months and 15 days remaining on his sentence.
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