The shortfall is driven not only by cuts in grants from Whitehall, but also by the escalating cost of adult social care - which has been predicted to outstrip available funding by £2.9 billion by 2020, said the Local Government Association.
New powers to hike council tax by 2% to pay for social care, announced by the Chancellor on Wednesday, will ease some of the pressure on town halls.
But even if all eligible councils take advantage of the full surcharge, they will rake in just £1.6 billion more by 2020 - at a cost of £96 a year to the average Band D council tax payer.
On top of the expected 1.99% annual rise in core council tax, this could see average Band D bills soar by £195 a household by the end of the decade without closing the black hole in budgets, according to LGA calculations.
And local government chiefs warned the change could widen the divide in service standards between comfortable and deprived areas, as councillors in poorer parts of the country hold back from rises which could be unaffordable to their residents.
The LGA calculated that, even where the 2% surcharge is applied, it will be worth more in wealthier areas with more expensive homes, where it will add 15% to councils’ budgets by 2020 compared to 5% in disadvantaged towns and cities.
LGA deputy chairman David Simmonds - Tory leader of Hillingdon in west London - warned this would exacerbate the “postcode lottery” in social care.
“If you are in one of the areas with a very low council tax base, unless you are somebody who physically cannot get out of bed, have a wash and feed yourself without someone coming round to your house, you are not going to get any help at all,” said Cllr Simmonds.
“In wealthier areas such as mine, we are going to be saying the council will provide you with a free burglar alarm, and if you need a heater the council will come round and provide one for you, because we are in a stronger position to do that.”
Mr Osborne told MPs that his settlement would leave councils in 2020 able to spend the same in cash terms as they do now.
He encouraged town halls to sell assets like shops or golf courses and dip into reserves to help pay for services.
But the LGA said that, after inflation is taken into account, councils in England will lose 24% of their core funding from central government by 2020, on top of a 40% cut between 2010 and 2015.
Even when cash they can raise from other sources, like council tax, is added in, spending power will fall by 6.7% in real terms, they said.
And LGA deputy chairwoman Sharon Taylor said that selling assets or raiding resources to pay for day-to-day spending would be “a fast-track to financial failure” for councils.
Ms Taylor - Labour leader of Stevenage Council - said there was no doubt that some authorities are “seriously under threat” of going bust because they will be unable to fund mandatory duties like adult social care and child protection, particularly as the full benefit from £1.5 billion injected by the Chancellor into the Better Care Fund will not filter through until the final years of the decade.
“You can cut out all the discretionary spending you like ... and push all that funding into delivering those key life-or-death services we provide, like safeguarding children and adult social care, and even if you do that, some authorities will still be under threat,” said Ms Taylor.
While civic leaders welcomed the ability to raise more money for cash-strapped social care systems, “we shouldn’t be relying on council tax payers to pick up the tab for a chronically underfunded social care system,” she added.
Jonathan Isaby, chief executive of the TaxPayers’ Alliance pressure group, said town halls should make savings rather than raise taxes.
“There are still savings to be had in town halls up and down the country,” said Mr Isaby.
“Taxpayers will want to see their local politicians prioritise services and root out waste, so all non-essential spending must be up for review.
“Furthermore, councils must put all of their assets under the microscope and look to share roles, services and back office functions with their neighbours to avoid tax hikes.”