As confirmed cases reach 16 in Tasmania, and over 1000 in Australia as a whole, the two islands have remained untouched to date.

Flinders Island mayor Annie Revie said the island would be particularly vulnerable if the virus did reach their shores.

"We have very, very small health facilities," she said.

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"We only have a six-bed acute hospital, and the aged care place is in the same building - and we have only two doctors.

"We've also got 17 per cent Aboriginal people, who are more at risk, and we also have a higher proportion than the rest of Tasmania of older people, and a higher percentage of people with acute conditions.

"So we have all these risks over here, and if people continued to come over here in high numbers and people got sick, then that would be ... very difficult."

Flinders Island Council are holding a meeting on Tuesday, where it will discuss future measures to limit the chances of the virus reaching the island, and to support the community in the interim.

That includes the possibility of taking steps to limit travel from Tasmania and the mainland.

"As an island, our chances of staying virus-free are probably not too bad, if we can contain things," Cr Revie said. "I can't really pre-empt that meeting, except to say that we will be discussing it."

The health hub on the island, called the Multi-Purpose Centre, has the capacity for testing.

At an Municipal Emergency Management Committee meeting on Monday, the MPC recommended that Flinders Island residents should only travel to Tasmania for essential medical reasons, including cancelling elective surgery.

It said the community - the island has a population just short of 900 residents - may be required to assist with caring and food drops.

Flinders Island businesses, many of which rely on tourism, have already lost upwards of $10,000 from the coronavirus fallout.

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There is also the possibility of Sharp Airlines, the primary airline servicing Flinders, reducing flights.

But Cr Revie said even if the number of flights per week decreases she does not expect major issues with goods reaching residents, especially considering the weekly freight which departs from Bridport.

Meanwhile, on King Island, mayor Julie Arnold said the council was not seeking to limit visitation from Tasmania - in fact, the opposite was the case.

"Come and visit the island now, there's certainly no restrictions coming in," she said.

"We're going along well, and we just want to make sure our businesses are as robust as they can be, because this really is a horrific time for them.

"We're social distancing, we're washing our hands, and we're looking after each other.

"Just don't come here to stock up toilet rolls."

The island's main airline, Rex (regional express) has not reduced flights to the island, keeping the number at seven per week.

On Friday it announced flights from Burnie to Melbourne return would be reduced from 27 a week to seven, as the airline struggles in the face of dramatically reduced demand.

However, Cr Arnold said the island would understand if King Island to Melbourne flights were also reduced in the future.

"Certainly I'm aware that the number of passengers on the Rex flights are diminishing," she said.

"Rex has always supported the island very strongly, and has been a part of our airport for 40 years."

King Island has a population of about 1,560.