Schools are following advice from the Ministry of Education about the impacts of the virus, while also making remote-learning contingency plans.
Otago Secondary Principals' Association president Linda Miller said that included ensuring there was enough space for social distancing at school assemblies, and having plans in place for school camps.
People sharing misinformation online was not helping, she said.
Facebook pages such as Dunedin News were a hotbed of unfounded rumours.
"It was really frustrating over the weekend," she said.
For example, one post said the ministry was calling schools about closures, which was not the case.
If a school pupil tested positive for the virus, that school might have to shut its doors for "a couple of days" and most would be well-placed to provide learning online.
The ministry was updating schools several times per week, and relevant information would then be passed on directly to parents and caregivers.
The University of Otago was working on contingency plans.
A spokeswoman said the University would take Ministry of Health and Ministry of Education advice as to whether to suspend lectures and move teaching online.
"In the meantime, we are making preparations to be able to do this quickly and with minimal disruption to students.
"All university academic departments are preparing to move teaching — both lectures and tutorials — online wherever possible and adapting course content to fit."
Work was also under way to consider how assessment and exams could be carried out if the University campuses were closed.
The university had also been planning how it would self-isolate students in halls of residence if necessary.
