Schools are remaining open after confirmation of two cases of coronavirus reaching the isles.
The council’s head of children’s services Helen Budge has been in contact with Scottish government officials.
It follows online posts of families self-isolating amid concerns over COVID-19.
However, health officials are said to have advised against self isolation, unless people are told otherwise, or are unwell.
The SIC has been “signposting” concerned parents to advice produced by Health Protection Scotland.
Schools manager Shona Thompson said advice over coronavirus was a “moving feast”, but said staff at Hayfield House had been in talks with public health representatives.
“We’ve been signposting folk to the advice that exists,” she said, highlighting a “very helpful document” from Health Protection Scotland.
“We’ve had conversations with staff in public health and what they are telling us is that you don’t self-isolate unless public health have told you to, or you are feeling unwell.
“We’re saying to folk, ‘this is the guidance, if you are unsure phone your GP and a clinician will phone you back’.”
She added health officials were also advising that that those potentially at risk would “have to have had the right amount of contact at the right time”.
“It is a moving feast as you can imagine”.
She added: “Helen [Budge] has had a conversation with the Scottish government and has passed our information onto them.
“We’ve been told that any school closures will be taken by the Scottish government – and it will be Scotland-wide.”
She added: “We’re not at that stage yet – we’re still in this ‘containment’ stage.”
It came as Prime Minister Boris Johnson held a Cobra meeting on Monday morning.
Meanwhile, NHS Shetland is working nationally and locally with partner organisations to actively
monitor two positive coronavirus (COVID-19) cases.
NHS Shetland Interim Medical Director Brian Chittick said: “While we recognise people may feel anxious about coronavirus, we would like to offer reassurance that we are taking all the necessary steps to try and contain the virus.”