Restricting travel from the mainland to Kangaroo Island was an option but one that required careful consideration, according to mayor Michael Pengilly.
The council in Robe, South Australia last week moved to restrict travel to protect its most vulnerable residents from coronavirus.
The mayor said ultimately it was a decision for the State Government and health officials and not the council.
"I've had a number of people approach me about it, but it's not really our decision and it's complex," Mr Pengilly said. "If we went with further restrictions, locals need to realise they would need to self isolate for two weeks each time they went to the mainland."
Federal Member Rebekha Sharkie is currently sitting in Parliament in Canberra and said any move to further travel restrictions on KI should be agreed at all levels.
"In times like these, being an island potentially has the benefit of a physical distance/isolation," Ms Sharkie said.
"Pushing for testing on Island was a critical first step and I'm glad the State Government has listened and acted in this regard.
"I will communicate with the Leon Bignell, the local State MP, Commissioner for KI and mayor in light of the recent announced SA shutdown. I think an agreed position either way would be the best way forward. "
Kangaroo Island Hospital staff have tested between two and six people a day for coronavirus at a special testing clinic.
KI Council CEO Greg Georgopoulos said the council was in contact with the State Government about various coronavirus related issues and options, but elected officials had not taken a position on travel restrictions.
The council was enacting a number of other measures including limiting public interaction and sending backroom staff to work at home, while still providing services to the community.
Work crews were going straight from home to their work sites.
It was business as usual on the farm for Mr Pengilly who on Monday, March 23 was crutching his flock with a team of shearers.
"Crutching goes on, regardless of COVID-19," he said. "Plenty of distance between shed crew. Sheep and dogs don't know about it!"
The SeaLink ferry service to Kangaroo Island meanwhile had already changed operating procedures based on advice from State heath officials.
Changes implemented already included asking people to stay in their cars during the crossing.
SA general manager Peter Abbott said the company was taking advice from the State Government and would work to implement any additional changes or restrictions if directed to do so.