Davidson was sentenced to one year in jail in September for using a fraudulent travel document.
Last week, 76 inmates escaped from a prison in Jambi province after floodwaters caused its fence to collapse.
Prabawa said the four men were believed to have escaped through a 50cm by 70cm (20 inch by 28 inch) hole found in a wall that connects to a 15 metre-long (49 ft) water tunnel heading towards a main street.
Said was seen by another inmate at 6:30am, so it's believed they escaped sometime after that.
Davidson is wanted in Perth, charged with possessing methamphetamine and cannabis.
He identified the four as Shaun Edward Davidson, 33, of Australia; Dimitar Nikolov Iliev, 43, of Bulgaria; Sayed Mohammed Said, 31, of India; and Tee Koko King bin Tee Kim Sai, 50, of Malaysia.
He had only 10 weeks still to serve of his sentence, according to Fairfax.
One of our very own was part of a 4-person group that escaped a Bali prison today.
He didn't attend his January 2016 court date and flew to Bali instead, which led to a warrant for his arrest being issued. They don't give you any food, they don't give you a bed. "It's pretty hard for some of the locals - if you don't have money to get food you don't eat".
Davidson had spent the year before he was apprehended boxing and partying.
Ngurah Rai airport immigration officer Mohamad Soleh told Fairfax Media in April past year that a report was made of a foreigner staying in Kuta who was suspected of overstaying his visa.
"We are investigating witnesses", he said.