Government departments have been told to outline potential spending cuts of up to 6% with the aim of saving up to £3.5bn by 2020.

Before the budget on 8 March, the chief secretary to the Treasury, David Gauke, announced that all Whitehall departments should submit ways to contribute to the government’s “efficiency review”.

Up to £1bn of the savings will be reinvested in “priority areas”, rather than being used to reduce the deficit, the Treasury announced, but did not give specific details.

The shadow chancellor, John McDonnell, accused the government of “sneaking out” the announcement late on Monday night and said it was proof that the chancellor, Philip Hammond, “represents more of the same Tory austerity”.

Departments have been asked to begin drawing up proposals based on two scenarios – one for making 3% spending cuts, and another for making 6%.

The cuts would come into force shortly after the UK leaves the EU, according to the two-year timetable for the article 50 process of withdrawal, and just before the next scheduled general election.

Gauke and the Cabinet Office minister Ben Gummer said the review was “part of an ongoing commitment to modern, flexible public services that deliver the best value for money for taxpayers”.

Core budgets for schools and the NHS will not come under review, the announcement said, and savings made by local government cuts will be used to ease pressures on social care. There was repeated criticism of the government throughout the winter for the crisis in social care, which NHS and council chiefs have said is chronically underfunded.

The Nato target of spending 2% of GDP on defence and legal commitments to spend 0.7% of GDP on international aid will also continue to be met, the Treasury said.

“We are committed to a modern, high-quality public sector that delivers the services people need in the most efficient way possible,” Gauke said. “There has been considerable progress, but there is further to go and the whole of government is working together to consider how we can live within our means while delivering maximum value for every pound of taxpayers’ money.”

McDonnell said the chancellor “urgently needs to explain why this announcement was made in the way it was and why after seven years of failed austerity he thinks more of the same will now work”.

He added: “For all Hammond’s talk of ‘resetting’ the Tory government’s economic policy last summer it is now beyond doubt that Hammond represents more of the same Tory austerity.

“Sneaking out an announcement asking departments to model cuts of 3% and 6% for 2019-20 is no way to manage public spending.”

