The research, conducted by the Local Government Association, found this number rose to almost two thirds by 2024/25.
The LGA, which surveyed 339 member councils, estimates the funding gap for local authorities in England could top £8bn by 2025.
Almost a fifth of councils (17 per cent) were not confident they will make the savings they need to in the current financial year.
An unprecedented rise in demand for children's services, adult social care and homelessness support has seen many councils forced to make in-year cuts to balance their budgets, reports the LGA.
Of the 141 councils that responded between March and June, 110 were able to indicate the year from which it was likely, on the basis of current funding, that they would no longer have enough funding to fulfil all of their statutory duties.
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The initial analysis has been released ahead of the forthcoming Spending Review, with fuller results due to be published later this year.
The report states: "Between 2010 and 2020, councils will have lost 60p out of every £1 they had from central government to run local services.
"The next Spending Review will be make or break for vital local services and securing the financial sustainability of councils must be the top priority."
It warned political uncertainty and an unresolved Brexit meant the chances of the government carrying out a three-year Spending Review this year looked increasingly unlikely.
Instead, councils should prepare themselves to face a one-year roll-over settlement, it said.
"Either way, councils urgently need some certainty about how local services will be funded next year so they can try and plan financially for next year.
"The LGA is calling for the next Prime Minister to prioritise local public services in the Spending Review and give councils certainty about future funding, business rates retention and the fair funding review."
The LGA is calling on the government to confirm funding for programmes such as the Better Care Fund, enabling more joined-up health and care services, and to provide councils with more freedom to set Council Tax levels.
Growing demand pressures particularly for areas such as children's services and special educational needs had contributed to a predicted funding gap of more than £3bn by next year, it says.
Lord Porter, LGA chairman, says the survey shows there was a "real risk to the future financial viability of some services and councils".
"Councils would normally have started their budget-setting planning process but remain completely in the dark about how much funding they will have next year.
"Communities relying on the vital local services that make a difference to their lives deserve better," he said.