It comes as Cook Islands Ministry of Health calls medical personnel, transport and immigration authorities, media and other stakeholders to a coronavirus response strategy meeting on Monday morning.
Chinese are the fastest-growing group of tourists in Cook Islands, accounting for 678 visitors last year.
Chinese authorities have revealed they are treating 1,287 patients with the virus – and the death toll has risen to 41. Authorities there warn the virus could mutate and spread, and have put three cities in lockdown.
The Melbourne patient is a man in his 50s who recently spent time in the city of Wuhan, where the deadly outbreak is believed to have originated in an illegal wildlife market.
New South Wales Health has confirmed five people in the state are being tested for coronavirus, but none of those cases have been confirmed.