The Local Government Association is urging Chancellor Philip Hammond to use the autumn Budget to allow councils to keep “every penny” of taxes raised locally.
It says an extra £1.3bn is needed “right now” to stabilise the “perilously fragile care provider market”.
The cost of caring for the elderly, vulnerable adults and children is set to soar from 41p in every pound spent on council tax to 56p by 2020, according to forecasts.
That will leave just 6p in every pound on collecting rubbish and recycling, 5p on roads and 2p on bus services.
Coun Claire Kober, chair of the LGA’s resources board, warned there was a “real and growing uncertainty” about how council services will be funded beyond 2020.
It comes after Government plans to allow councils to keep all of their business rates income were left in doubt after the Local Government Finance Bill, which was passing through parliament before the election, was not reintroduced in the Queen’s Speech.
Coun Kober said: “Local government in England faces a £5.8bn funding gap by 2020.
“Even if councils stopped filling potholes, maintaining parks and open spaces, closed all children’s centres, libraries, museums, leisure centres, turned off every street light and shut all discretionary bus routes they still would not have saved enough money to plug this gap in just two years.”
In addition to the surge in social care costs, almost half of councils in England will no longer receive any core central government funding in the form of the Revenue Support Grant by 2019/20.
Leader of Hull Council Coun Steve Brady said “strong leadership” was needed to find a cross-party consensus on an adult social care policy, which would survive changes in Government.
But he had “no faith in politicians as a whole” in finding a solution.
He said people couldn’t avoid paying, either through personal finances or through the tax system, adding: “The wealthier people have the ability – they may not like it – to pay for that care.
“But if you look at a place like Hull, I don’t think most people have the ability to provide much at all in terms of payments.”
Coun Michael Harrison, executive member for adult social care for North Yorkshire County Council, said the county already had an age profile five years ahead of everywhere else.
He said: “We are approaching (the point where) 43 per cent of everything we have to spend is on adult social care.
"We need an honest conversation (about paying for care). We were denied that at the election. I think the message was lost.
"We need to have an honest conversation about the realities of the costs and how we want to balance that through central, local government and private contributions.”