The Treasury has told most government departments to prepare "illustrative plans" to cut spending by 40% - as well as the expected 25% - within the month.
Education and defence have been given some protection, and must produce plans to cut 10% and 20%. International aid and health budgets are being protected.
The full 40% plans are unlikely to be implemented but will inform future decisions on cuts, the BBC understands.
Labour says the news will raise fears about the future of frontline services.
The government says cuts are needed to reduce the UK's £156bn annual budget deficit - the amount it must borrow each year to plug the gap between public income and spending.
However, public sector unions are threatening "co-ordinated industrial action" if ministers try to implement cuts as deep as 40%.
BBC political correspondent Gary O'Donoghue said ministers were told by the Treasury at a cabinet meeting in Bradford last Tuesday to draw up what are being called illustrative plans, and have them ready by the end of July.
Sources have stressed that Labour's pre-election plans had already implied cuts of 20% around Whitehall, he said.
Privately, they say that no department will actually face 40% cuts.
However, all departments have also been told to look at cutting their administration costs by a third, or even a half.
These are not final settlements - but they will allow ministers to see the concrete results of cutting budgets, our correspondent added.
Prior to the election, the Conservatives pledged to cut an additional £6bn above Labour's planned spending reductions.
Labour and the Liberal Democrats campaigned on the basis that to cut spending so deeply this year would threaten the economic recovery.
However, the Lib Dems - now in coalition with the Conservatives - have argued that since the election, the markets' loss of confidence in Europe's economies calls for swifter action.
A Labour Party spokesman said the talk of "swingeing, across the board cuts" would worry people that the government would not protect frontline public services and that tackling the deficit had to be done in a fair way.
"Already, we have seen the government showing an appetite for cuts and tax rises that particularly hit families on middle and modest incomes," he said.
"When it comes to departmental spending, the government should be willing to sacrifice some of its unworkable pet projects - like Swedish schools or political police commissioners - before making cuts that will damage the quality of the frontline services people rightly expect."
'Social consequences'
John Philpott, chief economist at the Chartered Institute for Personnel and Development, said the government had made clear that as well as cutting the fiscal deficit it wanted a "once in a generation reform of the public sector".
"Clearly it means the government has in its sights quite swingeing cuts," he said.
But he added that the option was only what "might" and not what "would" happen.
"If you delivered job cuts of that size it would not only hit back office functions, management and admin but it is inevitable it would be acting to the detriment of front-line services which would be politically unpopular," he said.
The Office for Budget Responsibility has forecast 610,000 public sector jobs will be lost by 2016, assuming departmental spending cuts of 25%. Mr Philpott said 40% cuts would put more like one million jobs at risk.
Mr Philpott said the "big economic debate" was over whether cuts would affect jobs in parts of the private sector closely dependent on the public sector.
"If the government's right then we can have cuts in the public sector and actually have a growing economy and more jobs.
"If the government is wrong, then unemployment is likely to rise, and the ecomony, if not facing a double-dip recession, is at risk of very slow growth for a long time," he said.
Industrial action
Austerity measures imposed in Greece after its economy was bailed out by the European Union and International Monetary Fund have been met with violent protests and strikes.
Public sector workers in Spain have also been on strike in protest at pay cuts averaging 5%.
Mark Serwotka, general secretary of the UK's Public and Commercial Services union, said there was "no economic case to put people on the dole queue".
"We are already drawing up plans with other public sector unions to ensure that if the government attacks our pensions, our jobs and public services, they will face resistance the like of which we haven't seen in this country for decades," he said.
"We will see not just co-ordinated industrial action by unions but campaigns in every community."
Chancellor George Osborne used his first budget to impose a tough package of spending cuts and tax rises, including raising VAT from 17.5% to 20% next January.
Other measures included a two-year pay freeze for public sector workers earning over £21,000, a three-year freeze in child benefit, a cap on housing benefit and a reduction in tax credits for families earning more than £40,000.
Mr Osborne said decisive action was required, given the dire state of the public finances.
