The director of public health at a major city council has instructed maintained schools not to open to more pupils from 1 June.
Sheffield City Council is also "strongly advising" non-maintained schools, such as academies, to delay wider reopenings, as it believes three of the government's five key tests will not be met on a local level by Monday.
It is the latest in a series of councils to go against the government's plan to reopen schools more widely from next week.
Snap survey: Only 1 in 5 councils back school openings on 1 June
A recent snapshot survey found that two-thirds of councils cannot guarantee schools will reopen to more children in Reception, Year 1 and Year 6 on 1 June.
In a statement shared online, Greg Fell, director of public health at Sheffield City Council, said he "does not feel assured" that, by Monday, he could be confident the rate of infection will be decreasing to manageable levels on a local basis, testing and tracing capacity will be in hand, and adjustments to the current measures will not risk a second peak of the virus.
He stressed that schools should not open more widely before 15 June – a fortnight after the government's chosen date.
Mr Fell said: "We will continue to monitor things closely and advise schools/settings when we do feel the time is right.
"For maintained schools, where the council is the employer, we are instructing schools not to increase the number of children attending more widely.
"For all other schools, academies, and childcare settings, we are strongly advising that they also delay their wider reopening.
"We will continue to review our advice and guidance to all schools and childcare settings but, we advise that neither should open more widely before 15 June, in order to allow for a full 14 days cycle of the test and trace systems."
Mr Fell said he feels confident that the government's first two key tests – to "protect the NHS’ ability to cope" and to "see a sustained and consistent fall in the daily death rates from Covid-19 so we are confident that we have moved beyond the peak" – are being met on a local level.
However, he said he "does not feel assured" the remaining three tests will be met in Sheffield by 1 June.
In relation to test three – to see "reliable data from Sage showing that the rate of infection is decreasing to manageable levels across the board" – Mr Fell advised, "We do not currently have access to data about community transmission of Covid-19 at a local level."
"We cannot be confident of the rate of transmission and spread of Covid-19 at a community level," he added.
Mr Fell said that test four – to "be confident that the range of operational challenges, including testing capacity and PPE, are in hand, with supply able to meet future demand" – is "crucial".
However, while he believes the council can supply PPE on a local basis, Mr Fell said "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," he said.
In relation to the final test – to "be confident that any adjustments to the current measures will not risk a second peak of infections that overwhelms the NHS" – Mr Fell said "we only want to increase the number of children in schools and settings when we are confident we have all the right systems in place locally and these are shown to be working and effective and that it is therefore as safe as possible for children and staff".
Abtisam Mohamed, Sheffield City Council's cabinet member for education and skills, added: "While nationally the government is requesting that schools and nurseries start to increase the number of pupils attending over the coming weeks, in light of the Sheffield public health advice, we do not yet feel assured that it is the right time and are advising our schools and nurseries to delay increasing numbers until 15 June 2020."
Education secretary Gavin Williamson said: "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 prime minister has announced that the government's five tests have been met, and based on all the evidence we will now move forward with our plan for a phased and cautious return of a limited number of pupils to primary schools and early years settings from Monday, and students in Years 10 and 12 two weeks later.
"This marks the first step in getting all children and young people back into classrooms so they can be with their friends and teachers again, and I'm enormously grateful for all the planning and preparation the sector has done in the lead up to welcoming these first pupils back."