Test and trace procedures were carried out and cases were then confirmed at St Helen's Primary School.
East Dunbartonshire Council's Ann Davie, Depute Chief Executive for Education, People and Business, revealed close contacts of the pupils affected are also self-isolating.
She confirmed there is "no evidence" of transmission within the primary school and added that "robust control measures" have been put in place at all schools in the local authority area.
NHS Greater Glasgow and Clyde said pupils and close contacts are being offered advice and support, and confirmed other St. Helen's staff and pupils can continue to attend as normal.
Ms Davie said: "We are fully supportive of Public Health at NHS Greater Glasgow and Clyde and have assisted with Test and Protect at St. Helen's Primary.
"Those individuals who have been identified as contacts are self-isolating at home.
"Robust control measures are in place at all our schools, and there is no evidence of transmission within the school.
"Schools are working with parents and carers to minimise the disruption to learning for those pupils who require to self-isolate and are providing reassurance to the whole school community."
An NHS Greater Glasgow and Clyde spokesperson said: "Our Test and Protect team is carrying out COVID-19 contact tracing linked to St Helen’s Primary School.
"To respect and maintain patient confidentiality, no further details will be released but we can confirm close contacts are being advised to self-isolate and being given advice and support.
"Apart from those who are identified as close contacts, all other staff and pupils can continue to attend as normal."