Another council has said its schools will remain shut to the majority of pupils on 1 June as local coronavirus cases continue to rise.
Last weekend the prime minister outlined plans to partially reopen schools for pupils in Nursery, Reception, and Years 1 and 6.
But Hartlepool Borough Council said that will not happen in its borough, stressing it will be using caution.
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The news follows confirmation that in Liverpool, schools will only be open to vulnerable pupils and the children of key workers on 1 June.
Individual headteachers have also stated their intention to go against the government’s plan.
In a statement, Hartlepool Borough Council said: "Given that coronavirus cases locally continue to rise, Hartlepool Borough Council has been working with schools and we have agreed they will not reopen on Monday 1 June.
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"Whilst we recognise the importance of schools reopening, we want to be absolutely clear that we will be taking a measured and cautious approach to this.
"We continue to work with schools to put in place appropriate measures to help keep children and staff safe when a phased reopening is possible."
Steve Reddy, director of children and young people's services at Liverpool City Council, said on Friday that he will write to all parents to tell them not to expect schools to reopen for everyone on 1 June.
His letter states: "Some parents have asked me when schools may fully reopen. This will vary from school to school.
"Each headteacher has to rigorously assess the risks of fully reopening for their particular school."
The Labour leader of Gateshead Council, Martin Gannon, said on Friday that his council's advice continued to be to "stay at home".
He said there was evidence that the rate of infection, measured through the R value, was greater than one in his borough.
And he said his views on the lockdown were echoed by his counterparts in Newcastle and Sunderland.
Mr Gannon told the BBC: "The national advice telling us the lockdown is over is frankly madness to Gateshead."
Patrick Roach, general secretary of teachers’ union NASUWT, said the government had not provided evidence that opening schools next month was safe.
"The government must now publish the scientific evidence it is relying on to claim that it will be safe for children to return to school," he said.