Sheffield Council is urging schools to stay shut because it says the Government's test and trace system is not ready.
The council says it has taken the decision after advice from Sheffield's Director of Public Health Greg Fell.
Sheffield's headteachers are now being told not to open their schools or increase numbers until June 15.
Cllr Abtisam Mohamed, Cabinet Member for Education and Skills, said a successfully operated test and trace programme is crucial to ensuring public safety.
But the council has been advised that the scheme may not be sufficiently established and robust by Monday, and so is delaying schools re-opening for two weeks to assess the effectiveness of the test and trace programme.
The National Education Union has welcomed the delayed re-opening, but said the council's decision should have been made earlier to avoid unnecessary anxiety and concern among teachers.
But the Education Secretary Gavin Williamson insists that scientific and medical advice says it is safe to re-open schools from Monday so long as social distancing is maintained after the Government's five tests were met.
The phased return to classes is due to start with children in nurseries, early-years settings, and Reception, Year 1 and Year 6.
We are acutely aware that many schools may have already put systems in place for increasing numbers of children prior to this date. "However, our priority must always be ensuring the safety of our staff and the children is not compromised when the numbers are increased."
Locally, we do not feel confident that the recently announced NHS ‘Test and Trace’ programme is sufficiently well established and robust enough to effectively manage any local outbreaks of COVID-19. "In Sheffield, we want more time for this programme to mobilise, be tested and demonstrate that it is working before we start to rely on it."
Throughout the COVID-19 pandemic our decisions have been based on the best scientific and medical advice, with the welfare of children and staff at the heart of all considerations.
The Government's reckless approach and last minute decision making should not be mirrored on a local level. "We trust and hope that in future Sheffield City Council make a proper and timely assessment of national advice and whether it is appropriate or safe for Sheffield to follow it without question."