Auckland Grammar School has joined calls for all schools to close immediately except for children under 14 who don't have anyone who can supervise them at home.
Headmaster Tim O'Connor said he spoke to Education Minister Chris Hipkins and Ministry of Education head Iona Holsted over the weekend to urge them to take the issue to today's Cabinet meeting, which is due to end at 1.30pm.
He said his school and some others would move to online teaching anyway if the Government doesn't decide to take the same decision for all schools.
"We are imploring that they make a decision to close schools across the country," he said.
"We would prefer that they make those decisions for the benefit of the whole country and all schools, but if that can't be done then I guess individual schools will have to start making calls to do what they think is best."
The Teaching Council and former Chief Science Adviser Sir Peter Gluckman have both urged the Government today to close all schools and early childhood centres until the end of the April school holidays on April 28.
However, Principals' Federation president Perry Rush said the Government should stick to its stated policy of shutting schools only if there is a widespread community outbreak of the coronavirus.
"We think it's very important that teachers and principals and schools are not forced into panic, that we are measured, and that we are careful on how we move forward," he said.
He said 25 to 40 per cent of students stayed home today anyway out of fear of the virus.
A group of parents from Auckland Grammar and other Auckland schools have written to Cabinet ministers noting that the World Health Organisation now lists New Zealand as having "local transmission" of the coronavirus - not just imported cases.
"Some parents have been confirmed to have contracted Covid-19, their sons and/or daughters are regarded as close contact but by definition they were not being tested for the virus, as a result some of these students have now been infected as well," the parents said.
The Ministry of Health's policy is that people are only tested for the virus if they have a history of overseas travel, or a close contact with a confirmed, probably or suspected case, and have symptoms.
O'Connor said schools were struggling to keep operating normally after the Government decision on Saturday that people over 70 or with compromised immunity should stay at home, because relief teachers are "typically 65 and over".
"We are down by 20 staff today - teachers over 70 or whose immunity has been compromised in some manner and just feel the need to be in self-isolation, and I'm aware that some schools are facing an even higher level than we are," he said.
"But also we have other bodies who are out there imploring closure. The Teaching Council has said that today, Sir Peter Gluckman has said that today, and that follows the sense that we are feeling within our communities, and we now have four schools in Aucklands who are closed.
"You start to appreciate that things are moving rapidly and we would like to be acting in a conscious manner for safety rather than continuing to be cautious."
• Coronavirus: Four Auckland schools linked to Covid-19 in one day
• Coronavirus: Pukekohe Intermediate School to close on Monday after pupil tested
• Coronavirus: Teachers call on Prime Minister Jacinda Ardern to shut down schools now
• Coronavirus: Ministry of Education reveals factors that will close schools, bring holidays forward
Auckland Grammar has already experimented last week by sending students in Years 9 to 11 home to work online for one day at each year level. Today Years 10 and 11 are both at home.
Students under 14 were still able to come to school if they did not have anyone to supervise them at home, or did not have internet or devices, but fewer than 1 per cent did so.
O'Connor said any national decision to move learning online should also allow students to still come to school if they did not have anyone to look after them at home or if they did not have suitable computer devices or home internet.
He said the decision should be made for the next five weeks until the end of the April school holidays, and should then be reassessed.