Headteachers have warned that the idea of Ofsted beginning full inspections again next term is now "unthinkable".
They argue that the disruption caused by rising levels of Covid-19 infection will make it impossible to carry out school inspections in a "meaningful" way.
And their case has been bolstered by yesterday's news, revealed by Tes, that a London primary had to close for a week after a visit from an Ofsted inspector who tested positive for Covid.
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Geoff Barton, general secretary of the Association of School and College Leaders, argues that Ofsted should not now return until September next year.
But when asked by Tes, both Ofsted and the Department for Education said the plan was still for routine inspections to start again in January next year, although this date was still under review.
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Mr Barton, who has already called for this term's Ofsted school "visits" to go online, questioned why a final decision has not yet been made.
"It is frustrating that there is still no decision on this issue and schools need clarity," he said.
"It is extremely unlikely that the Covid situation will improve to any extent over the coming weeks, and it seems pretty straightforward to make a call now that routine inspections will not resume in January in the light of the current circumstances.
"Our view is that the likelihood of ongoing disruption, and the need for schools to be able to focus on supporting children in these turbulent circumstances, means that routine inspections should be suspended for the remainder of this academic year."
Mr Barton added: "Ofsted and ministers must realise that resuming routine inspections in January is clearly unthinkable given rising Covid infection rates and widespread disruption as pupils and staff have to self-isolate.
"We cannot see how routine inspections could be carried out in a meaningful and consistent manner amidst this level of turbulence.
"And, at a time when schools are already under huge pressure managing control measures and endeavouring to stay open, the last thing they need is this additional demand.
"There is also surely a public health consideration in avoiding a situation in which inspectors are going in and out of schools, particularly in high-risk areas."
Tes asked both the DfE and Ofsted when they will make a final decision on whether to go ahead with full inspections in January; what, if any, criteria they are using to make this decision; and what Ofsted will do in the intervening period if it is decided to defer inspections further beyond January 2021.
None of these questions have been answered by either organisation.
When asked if any other organisation was involved in the decision, Ofsted said that it was working closely with the DfE on this and that the date for a return to inspections was January next year.
A DfE spokesperson said: "While routine inspections will remain suspended for the autumn term, Ofsted is visiting schools to discuss how they are managing the return to education of all their pupils.
"We intend to restart routine inspections from January 2021. However, as we always said, this date is being kept under review.
"When inspections do restart, inspectors will be sensitive to schools' circumstances and the enormous challenges presented by Covid-19."
Mr Barton said that stopping inspection until next September would "give school leaders and their staff some badly needed reassurance and certainty about what to expect".
"It would demonstrate that the government and Ofsted recognise the circumstances and are being supportive," he added.
"It is the least that schools deserve after the huge amount of work they have put into full reopening and managing this incredibly difficult situation."