Council tax rises due to come into force from April will not be sufficient to avoid deep cuts to services including road repair, parks, children’s centres, leisure centres and libraries, local government leaders have warned.

The Local Government Association (LGA), which represents councils across the country, said authorities will have to continue cutbacks to essential services to plug growing shortfalls in adult social care.

The body said it believes 147 out of 151 authorities responsible for social care will raise £540m more over the next financial year through a local precept, but on council tax this will be swallowed up by the extra cost of paying the national living wage to care workers.

They are allowed to raise council tax by almost 2% in 2017/18 to fund local services and a further 3% specifically for social care. Any rises beyond that would mean the council having to seek permission through a local referendum. The LGA said 108 councils were considering a 3% social care precept and 39 were looking at a 2% precept for social care.

It said this would not be sufficient to avoid cuts elsewhere, and councils would have to divert money currently allocated for other local services, including filling potholes, maintaining parks and green spaces and running children’s centres, leisure centres and libraries.

Lord Porter, the Conservative chairman of the LGA, said services supporting very vulnerable people are “at breaking point and many councils are increasingly unable to turn down the chance to raise desperately needed money for social care and other local services next year.

“But extra council tax income will not bring in anywhere near enough money to alleviate the growing pressure on social care both now and in the future. The social care precept raises different amounts of money in different parts of the country.

“Social care faces a funding gap of at least £2.6bn by 2020. It cannot be left to council taxpayers alone to try and fix this crisis.”

The prime minister has stood firm against demands from councils and medics for more money for social care, despite warnings that stretched services are putting unsustainable pressure on the NHS. Instead, she has pledged to seek a “long-term solution” to the challenge of funding social care for older people, and said local authorities would be allowed to bring forward increases in council tax to ease the pressures.

Philip Hammond, the chancellor, is likely to come under renewed pressure to help councils and the NHS at the budget next month.