The Local Government Association estimates it could cost councils at least £330m in 2016-17 to cover increased contract costs.
It wants £700m of new government funding earmarked for social care by 2020 to be brought forward.
The government said it was offering more financial support to councils.
The government is allowing councils to raise council tax by 2% to help meet the rising costs of social care, under plans announced in the 2015 Autumn Statement.
According to Local Government Association  (LGA) figures, this could bring in about £372m in 2016-17.
But it warns this could be wiped out by meeting the costs of the new £7.20 National Living Wage, for those aged over 25, which comes into force in April.
The LGA says the increased costs of contracts for home care and residential care providers could cost councils a minimum of £330m in 2016-17.
"The cost of implementing it will significantly add to the growing pressure on services caring for the elderly and disabled which are already at breaking point," said councillor Izzi Seccombe, the LGA's community wellbeing spokeswoman.
"A lack of funding is already leading to providers pulling out of the publicly-funded care market and shifting their attention towards people who are able to fully fund their own care.
"We know that care home and domiciliary care providers cannot be squeezed much further."
The government said care workers would benefit from the National Living Wage, and added that it was making up to £3.5bn available for councils to meet social care costs.
A spokesman said: "The new National Living Wage (NLW) is an essential part of moving to a higher wage, lower tax, lower welfare society, where work always pays and the majority of households are better off.
"The NLW will benefit hundreds of thousands of care workers for the work they do and we are offering more financial support to councils to invest in that work - with access to up to £3.5bn to spend on social care."
Care services for the elderly and disabled could hit "breaking point" when the new National Living Wage comes into effect, councils say.