One Cabinet minister told The Telegraph: "We have got to make sure this economic downturn is V-shaped and not L-shaped. We should be beginning to release the things that can be released – so primary schools should reopen and so should non-essential shops.

"If you can go into Sainsbury's to buy non-essential items while observing social distancing rules, why can you not do that in other shops?"

Dismissing fears that parents might be fearful of term time resuming for fear of their children inadvertently spreading the virus to grandparents, the minister added: "It looks like elderly and vulnerable people are going to be kept in self-isolation for six months rather than three."

Sir Iain Duncan Smith, the former Conservative Party leader, said: "Schools are important because they enable parents to go back to work, particularly primary schools because those are the children who are too young to be left at home alone. Reopening primary schools is the key to unlocking labour."

Working in tandem with the Treasury, Downing Street is said to be formulating plans for a gradual easing of the restrictions on a sector-by-sector basis.

But one Tory source said Gavin Williamson, the Education Secretary, had been "flip-flopping" on the issue of re-opening schools, adding: "He's not really a decision-maker. One minute he seems to believe schools should reopen, but the next he's being overly cautious.

"It's hard to work out where he stands on it from one day to the next, but he is having to balance a lot of competing interests."

On Tuesday, one teachers' union wrote to the Government to say its members were "disturbed" by suggestions that schools could reopen.

The letter from the National Education Union read: "Given that, in reopening schools and colleges, you would be asking our members to take an increased risk, we believe they have a right to understand fully how any such proposal belongs within an overall Government strategy to defeat the virus."

When Boris Johnson announced the lockdown on March 23, prompting GCSE and A-Level exams to be cancelled, it had been suggested that schools may not reopen until after the summer holidays in September.

But calls for a return sooner than that have been fuelled by research by University College London which found that keeping pupils off has had little impact, even with other lockdown measures.

Recent modelling studies of Covid-19 predict that school closures alone would prevent only two to four per cent of deaths, many fewer than other social distancing interventions.

The Government had expected 20 per cent of children to stay in schools after the lockdown was put in place, but actually just two per cent of vulnerable pupils and the children of key workers have carried on attending.

A Number 10 insider said Mr Johnson's gradual return to work following a week in intensive care, along with his chief adviser, Dominic Cummings, following his own coronavirus isolation "would have an impact on the dynamic of the lockdown".

The insider said: "Dom is worried about lifting the restrictions too early and Britain suffering a second Covid-19 wave in the autumn which could cause even greater economic damage."