Twechar Primary, Aileymill Primary in Greenock, St Monica’s Primary in the Pollock area of Glasgow and Gartconnor Primary in Kirkintilloch are all targeted.
Also on the list is Knightswood Secondary School near Glasgow.
The Daily Record includes that two children tested positive in the Aileymill Primary and that two-year-olds were asked to self-isolate.
A small group of employees at UK-based McKesson healthcare company All About Health on Polmadie Road were also recorded.
Public health teams have been formed and contact lists for the testing and protection team have been identified.
Close contacts are asked to self-isolate, but all other staff can continue to attend as normal.
The health council insisted that there is no risk associated with the handling of drugs in the company.
Apart from those who have been in close contact with a positive case, all other school staff and students have been informed that they can continue to attend normally.
The health council says there is no risk to the community at large.
The public has been urged to remain vigilant.
There is no need to request a test if you do not experience any symptoms, but anyone with one or more symptoms should immediately self-isolate and order a test.
It comes after Prime Minister Nicola Sturgeon announced an increase of 160 coronavirus cases earlier today.