The instruction - to be confirmed in a letter from Treasury Chief Secretary Danny Alexander to Cabinet ministers - is the latest step in preparing for what is set to be the toughest spending review since the Second World War.
Officials last night stressed that what was being prepared was "initial planning assumptions" and that the cuts on that scale would not actually be implemented in the final settlements in the autumn.
Nevertheless, the figures underline just how severe the review - which is due to be announced in October - will be.
The Office for Budget Responsibility forecast that 600,000 public sector jobs will be lost as the coalition Government tackles the deficit.
In his emergency Budget last month, Mr Osborne warned that departments faced cuts averaging 25% over the next four years - apart from health and overseas aid whose budgets were ring-fenced.
However he also indicated that defence and education would receive favourable treatment.
The independent Institute for Fiscal Studies estimated that this could leave other departments facing cuts of 33%.
Mr Osborne has now instructed the Ministry of Defence and the Department for Education to draw up plans showing the impact that budget cuts of 10% and 20% would have.
The rest of Whitehall was told to prepare plans for cuts of 25% and 40%.
Ministers were informed of the latest preparations at the weekly meeting of the Cabinet on Tuesday, which was held for the first time since the new Government came to power outside London, in Bradford.
A Treasury spokesman said: "We are determined to tackle the record budget deficit in order to keep rates lower for longer, protect jobs, and maintain the quality of essential public services.
"The Cabinet has been briefed on the planning assumptions that their departments should use for the initial phase of the spending review.
"These planning assumptions are not final settlements, and do not commit the Treasury or departments to final settlements.
"These assumptions will be negotiated so that we both tackle the deficit and support the freer, fairer and more responsible Britain we want to see."
Labour leadership contender Ed Balls said the latest details of the Government plans would "send a chill down the spines" of public sector workers and people who rely on public services.
"The Tories are trying to whip up a sense of crisis and inevitability to soften the ground and provide cover for what they've always wanted to do," he said.
"Their purely ideological programme is to roll back the state, slash public services and dismantle the welfare state.
"It's extraordinary that the bulk of the Liberal Democrat party have been fooled into going along with this right-wing political project.
"As economists have warned, these plans risk a double-dip recession and will hit those on modest incomes hardest."
Shadow home secretary Alan Johnson warned that the scale of the proposed cuts meant policing would inevitably suffer.
"The idea that you can cut the budget in the Home Office by 25% - 33% according to the Institute for Fiscal Studies, 40% we're seeing in this morning's story - that you can do that to a budget of 10 billion and you won't affect frontline policing is a fantasy land," he told BBC1's The Andrew Marr Show.
He said the Government's failure to protect the police budget - as Labour had been planning to do - meant counter-terrorism would inevitably be hit.
"There is no way round it I am afraid. It's very labour-intensive and there is a lot of technology involved as well, but it is crucial to the safety of this country," he said.
Transport Secretary Philip Hammond - who was the Conservative shadow chief Treasury secretary in opposition - acknowledged that achieving the cuts would be difficult, but he insisted it was necessary.
"Of course it will be very challenging to take out even 25% - certainly going beyond 25% gets more and more challenging, but that doesn't mean that we shouldn't look at each line of our budgets," he told The Andrew Marr Show.
He warned that spending of future road and rail projects would be affected.
"We are going to have to look at all of these different projects, we are going to have to prioritise aggressively, do the things which deliver the most value, which deliver the most support for economic growth," he said.