It's "just essential" that contact tracing resumes for primary schools, a teachers' union says.
INTO also wants to see the rules around mask-wearing for younger children reviewed.
The recent increase in COVID-19 cases has led to doubts that all aspects of this week's planned reopening will go ahead as previously planned.
It's also raised concerns about the situation in schools, with some already reporting significant outbreaks.
The vast majority of primary school children aren't vaccinated, as no vaccine has yet been approved for children under the age of 12.
Last month, the rules around routine contact tracing in schools changed, and the process is no longer carried out for close contacts within schools.
INTO President Joe McKeown told The Hard Shoulder it's now clear that change happened too early.
He said: “We had that system in place - it was there at the beginning of this year. We always felt it was premature to remove it when we hadn’t had sufficient data.
“That support has been removed from schools, and boards of managements… are left making their own decisions when they see cases exploding.
“It is important that children who are well come to school. But what we have seen happen in the last number of weeks is because parents don’t have information and are hearing things from word of mouth… they’re removing their children from school anyway.
“Just pretending it’s not there is certainly not a solution to the problem.”
He said his union objected to the change in the first place, but they now "absolutely believe" the Department of Education should go to the HSE and ask for contact tracing to be reinstated.
He said: “There was very scant evidence in the first three weeks of September. What we know very clearly is that COVID-19 cases rising in general - in the absence of hard evidence [around schools], we need to take as many precautions as we possibly can.
"We cannot continue with a situation where we just cross our fingers and toes and hope everything is OK. We really need to do practical things to keep our schools safe.”
Antigen testing and masks
Mr McKeown said mask-wearing and antigen testing in schools would be key "low-cost interventions" that should be acted on immediately.
He noted that the Chief Scientific Officer Professor Mark Ferguson proposed a pilot antigen testing scheme for schools earlier this year, but it simply hasn't happened yet.
INTO now believes that pilot should take place in a small number of schools so experts can then properly analyse the data.
Meanwhile, Mr McKeown says his union acknowledges that mask-wearing could be problematic for younger children.
However, he observed: “We certainly think for older children in primary schools, it should be given serious consideration.”
Under current rules, masks are only required in secondary school classes.
Mr McKeown noted that means a 12-year-old does not have to wear a mask in sixth class of primary school, but other 12-year-olds in their first year of secondary school do.
INTO now wants to see an "urgent review of the age restriction on the use of face coverings".