The National Education Union (NEU), the largest teaching union in Europe, has written to the watchdog warning that schools are having to manage “parental and student anxiety”.
They are also making special arrangements for vulnerable pupils and staff, and planning work in the event of closures, the letter from the NEU says.
The government was facing mounting calls to shut educational establishments down on Sunday, when “close the schools” was trending on Twitter in the UK.
It comes after the Association of School and College Leaders (ASCL) called on the government to immediately halt all Ofsted inspections on Saturday.
The watchdog has said it will defer visits to any school or college affected by the virus.
But the letter, from the NEU’s general secretaries, said: “We are very concerned, given the serious challenges that Covid-19 poses for schools, that Ofsted is still planning to conduct inspections.
“We fail to understand that, in light of the government’s decision to move to the ‘delay’ phase, Ofsted has revised its deferral policy to defer inspections on providers that are ‘affected’.
“We have to ask the question – can you name any school in England which has not been affected by Covid-19?”
It adds: “Even if there are no reported cases in schools, all leaders and staff are highly alert and responding to a range of pressing concerns and issues about the management of Covid-19, all of which involve changes to school routines and an intensification of already excessive workloads.”
Addressing hundreds of headteachers at its annual conference in Birmingham, Geoff Barton, general secretary of ASCL, said it was not a time for “business as usual” but a time of “national emergency”.
Mr Barton also called on Ofsted to show it understands the “extraordinary pressures” on schools. One school leader, he said, currently has 15 members of staff off because of coronavirus.
“It simply cannot be right that schools and colleges are judged in a high-stakes manner in these extraordinary circumstances,” he said.
Education secretary Gavin Williamson was due to meet with headteachers’ unions on Monday to discuss the variety of concerns that school leaders have over coronavirus.
Addressing the ASCL conference, Sean Harford, Ofsted’s national director of education, said: “We will be proactive in asking schools and colleges if they want to request deferral because of coronavirus and clearly we will look very favourably on all such requests.”
The chief inspector has asked to be “personally involved” in any decision not to defer, Mr Harford said.
Speaking to the media, Mr Harford added Ofsted was “not able to say that we’re ceasing inspections, that’s something for the government to decide” and would need “legislative change” to do so.
An Ofsted spokesperson said: “Ofsted is working closely with the DfE and monitoring the situation day by day. In line with current government advice, normal inspection activity should be maintained as far as possible.
“However, the spread of the virus presents serious challenges for providers that are affected.
“In line with the potential for this to be an ‘exceptional circumstance’, we will carry out a risk-assessment with the providers when we announce the inspection. This will give the leaders/managers of providers the opportunity to inform Ofsted about any current coronavirus impact on their provision.
“Using this information, we will make an assessment and a deferral decision, as appropriate. When inspections go ahead, inspectors will be sensitive to increases in user absence or absence of key staff, and will reflect this in the inspection report.”
On Friday, Mr Williamson told heads: “At present we are clear that the best course of action is to keep schools open unless you are advised to close by Public Health England.”