Gavin Williamson said today he could not give a date for school reopenings because five tests would have to be met first.
The education secretary, who led the daily Downing Street coronavirus briefing, said: "People are anxious to know when we're going to relax restrictions, when schools are likely to be fully back and open again.
"Of course, I want nothing more than to see schools back, get them back to normal, make sure the children are sat around, learning, and experiencing the joy of being at school.
"But I can't give you a date. Because before we do, we need to meet five tests."
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Outlining those five tests, Mr Williamson said: "First we must protect the NHS' ability to cope, and be sure that it can continue to provide critical care and specialist treatment right across the whole of the United Kingdom.
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"Second, we need to see daily death rates from coronavirus coming down.
"Third, we need to have reliable data that shows the rate of infection is decreasing to manageable levels.
"Four, we need to be confident that testing capacity and PPE [personal protective equipment] is being managed, with supply able to meet not just today's demand but future demand.
"And fifth, and perhaps most crucially, we need to be confident that any changes we do make will not risk a second peak of infections.
"When we can be sure that we have met these five essential points, we can think about getting children into schools again, learning, mastering new ideas and being with their friends once more."
His comments came on the day that one newspaper had reported that schools could reopen as early as 11 May, as part of a three-phase plan to lift lockdown measures in the UK.
Mr Williamson also rejected the idea contained in the same report that one option could involve schools opening during the summer holidays.
"There are currently no plans to have schools to open over the summer holiday and we haven't been working on plans to have them open over the summer holiday," he said.
He thanked those in the education sector for keeping schools and nurseries in operation during the lockdown.
He said: "I'm profoundly grateful for the vital roles they are playing in our communities."