Michael Gove has backed a public debate over the lifting of lockdown measures as he warned pubs and restaurants would be among the last to see restrictions eased.
But he played down the idea schools could reopen in the middle of next month as part of a traffic light system.
Ministers are coming under increasing pressure from senior Conservatives and others to level with the public, amid accusations they are treating people like children.
Opposition parties have also called on the government to set out a strategy for how it will decide how and when to lift the coronavirus lockdown, which will see Britons asked to stay at home for at least another fortnight.
Mr Gove told Sky Newss Sophy Ridge on Sunday: “It is the case that we are looking at all of the evidence, but we have set some tests which need to be passed before we can think of easing restrictions in this lockdown.”
He added: “It is entirely understandable, of course, that there should be a public debate about how we approach these difficult choices.”
On schools he told BBC Ones The Andrew Marr Show: “We have stressed that the reporting in todays newspapers that schools will reopen on May 11, that is not true, we have not made that decision.”
In a blow to the UKs pubs and restaurants, set to lose billions of pounds because of the pandemic, Mr Gove said: “Areas of hospitality will be among the last to exit the lockdown – yes, that is true.”
Denmarks ambassador to the UK also called on ministers to be honest with the public about exit strategies.
Lars Thuesen told Sophy Ridge on Sunday on Sky News the Danish government had been upfront about how it would ease its lockdown.
The Danish government had agreed a sort-of pact with its people, he said.
He said: “It does work. Before Easter, the Prime Minister told the Danes that it would be possible to ease some of the restrictions if they behaved during Easter time, during these wonderful summer temperatures, if they behaved responsibly, if they continued what they had done the first weeks, then we would be able to open up again. So it was sort of a pact between the government and the population.”