Lord Porter, chairman of the Local Government Association, told Sky News' All Out Politics that councils should be able to raise the tax by as much as they like to fund services.
He called for the current cap to be removed.
But Lord Porter said funding care for the elderly was a "national problem" and "that part of funding should come from national government".
He was speaking after the Government relaxed its council tax cap, meaning councils will be allowed to raise the charge by up to 5.99% next year.
Communities Secretary Sajid Javid told MPs this will give local authorities "the independence they need to help relieve pressure on local services" while "recognising the need to keep spending under control".
The LGA, however, has warned Downing Street the move will raise just £250m a year, at a time when social care service leaders estimate that £6bn has been cut from local care budgets since 2010.
Lord Porter said he thinks most councils will end up increasing the tax by less than 3%, but warned his members were "squeezing services to the bone" as a result of funding cuts and being asked to "get a quart out of a pint pot".
On how adult social care should be paid for, he told Sky News: "We've got to tax people somehow.
"It doesn't have to be income tax … property tax … all sorts of things. We pay fuel tax, we pay stamp duty, we pay road tax, people are paying tax all over the place.
"I don't care which pot they take it from, the Treasury needs to get it out of the Treasury and get it to my members.
"Nobody likes paying tax. But if we're genuinely a caring society, we have to be prepared to pay the price for that."
Mr Javid made his announcement as part his provisional local government funding settlement for 2018/19.
Previously, councils had to hold a referendum if they wanted to put up bills by more than 2%. This has now been increased to 3%.
Coupled with the additional 3% precept given to authorities with social care responsibilities, this gives councils the ability to increase charges by up to 5.99% next April without putting it to residents in a vote.
A Department for Communities and Local Government spokesman said: "This settlement strikes a balance between giving councils the ability to make decisions to meet pressures and ensure that our most vulnerable in society get the support they need while protecting residents against excessive council tax bill rises."
Labour's Andrew Gwynne slammed the package as "piecemeal" and said the Government did not have a "sustainable plan" for the future of social care.
Meanwhile, the TaxPayers' Alliance said it "beggars belief" that authorities are being allowed to hike council tax at a time when most people's pay packets are rising at levels below inflation.