A number of further COVID-19 cases have been found at Calderglen High after parents were initially informed of positive test last week.
NHS Lanarkshire have sent a second letter to parents after a pupil previously tested postitive.
It is understood the latest cases over the September weekend also relate to pupils.
The letter reads: "NHS Lanarkshire's test and protect team have been carrying out contact tracing after a number of individuals at Calderglen High tested positive for COVID-19 over the weekend break. These individuals are currently self-isolating at home.
"NHS Lanarkshire has been working closely with South Lanarkshire Council and is in the process of contacting close contacts."
But the letter, dated September 29, adds that following a review, there is "no evidence currently to suggest that there is transmission in the school" and that the risk to the wider school community is low. Therefore the school will remain open.
A letter, dated yesterday (September 30), has also been send to parents and carers at Duncanrig Secondary saying there has been a further case found there.
A SLC spokesman said: "We continue to work closely with NHS Lanarkshire, which leads on the handling of all cases where COVID-19 is suspected or confirmed. Appropriate action has been taken and information on this has been relayed to all parents, carers and staff.
"We would stress that there is currently no evidence of transmission in the schools and the risk to the wider school community has currently been assessed as low. The schools will therefore remain open.”
There have been a number of schools in East Kilbride and Strathaven - both primary and secondary - where staff or pupils have tested positive for COVID. Five staff members at Greenhills Primary were among them.
