The HSE has estimated that between 10,000 and 12,000 children are restricting their movements as a result of being designated a close contact of a Covid-19 case.
The calculation is based on around 700 primary schools and 500 secondary schools reporting a Covid-19 case.
Each primary school case would have an average of 15 close contacts and around four in the case of secondary schools.
Niamh O'Beirne, the head of HSE Test and Tracing, told RTÉ News that it will take some days to test the children and get the results.
A fuller picture of the situation in schools is expected at the end of the week.
Ms O'Beirne said the vast majority of infections are expected to have come from the community, not school transmission.
She also said that swabbing centres saw one of the busiest days this year with around 20,000 swabs taken.
In Northern Ireland, Sinn Féin has called for the Stormont assembly to be recalled to deal with the rising number of pupils who are missing from school because they have tested positive or been in close contact with someone who has contracted Covid-19.
Some teachers have voiced concern over the number of pupils who are absent less than two weeks into the new school term. In one instance, the principal of Larne High School reported that more than half of the pupils have been told to stay at home.
Education Minister Michelle McIlveen said that the current situation in schools is "a challenge".
She told the BBC: "I've spoken to the health minister this morning and I plan to do that again as we go through the day, just to see what we can do in order to try to assist schools.
"Our main focus is about ensuring that our young people remain in education and that we don't have prolonged lengths of time when they are away from school."
Additional reporting by PA
