Wealthier homeowners should be asked to make a voluntary payment of up to £30,000 for their care needs in old age, a new report argues.
The Centre for Policy Studies proposes a system in which everyone receives a state-funded weekly care payment.
Those able to downsize or release equity from their homes would also be encouraged to contribute more to plug the current funding gap.
But critics say it would not be enough to address the £7bn shortfall.
Labour shadow chancellor John McDonnell called on the government to reject the plan, which he said would "punish older people with a tax on getting old".
But former work and pensions secretary Damian Green, the report's author, told the BBC that if "just a sliver" of the equity tied up in property could be released, it would inject substantial amounts of funding.
Social care for the elderly covers non-medical needs such as help with washing, dressing and eating.
Everyone with more than £23,250 has to pay for support. Below that threshold, they contribute to the cost - with the amount paid based on means-testing of both savings and income.
Attempts by successive governments to reform provision in England have foundered amid political disagreements and concerns over the financial costs involved.
The Conservatives dropped plans in 2017 to make people receiving care at home liable for the full cost if they were worth at least £100,000 following a political outcry.
Theresa May was accused of trying to introduce a "dementia tax" by charities and pensioner groups who said people would no longer be able to pass their homes down to their children if property values were taken into account when calculating care costs.
BBC health correspondent Nick Triggle said the failure to reform had come at a price, with only about one in five of those who need help now getting it from the state - the rest rely on friends and family, pay for care themselves or, increasingly, go without the support they need.
How previous care plans bit the dust
Dilnot's care cap: In 2011, economist Andrew Dilnot was asked to lead a review of the system. The coalition government accepted the principle of his main recommendation, a maximum £50,000 cap on individuals' lifetime care costs, although David Cameron later set a higher figure of £72,500. The Tories dropped the idea in 2017, saying they would bring forward their own blueprint - which is as yet unpublished.
Conservatives' "dementia tax": Theresa May's 2017 election manifesto promised a controversial shake-up of care funding. Under the plan, the value of an individual's home was to be taken into account when assessing both domestic and residential care costs. Critics said it left people facing unlimited costs. The PM quickly did a u-turn, saying costs would be capped after all - although the plans were quickly shelved after she lost their majority.
Labour's "death tax": In the run-up to the 2010 election, the then Labour government proposed charging a 10% levy on the estates of deceased people to pay for care costs incurred during their lifetime. It was championed by the then health secretary Andy Burnham, now Greater Manchester Mayor. Amid a political backlash, the plans were dropped.
In its new report, the Centre for Policy Studies - a conservative think tank - said the current system was "financially and politically unsustainable".
It says taxes will need to go up to address the rising costs of care, with one option to plug the gap in the short-term being to ask over-50s to pay more National Insurance contributions.
Other possibilities for a quick fix include taxing the winter fuel allowance or allocating more money in this autumn's Spending Review.
Longer-term though, Mr Green argues, a radical overhaul is needed, bringing in a Universal Care payment each week for everyone regardless of their wealth.
This would be similar to the state pension allowance and be paid for out of taxes.
On top of this, extra funding would then be generated through an annual care supplement, in which homeowners could choose to make voluntary contributions of either £10,000, £20,000 or £30,000, either through downsizing or releasing equity.
Mr Green told Radio 4's Today that "many millions" of people would be able to buy what he described as an "insurance policy" although, unlike previous proposals, they would not be forced to.
"Given you have this huge raft of wealth, particularly in housing, a sliver of that used for social care would buy peace of mind for people in their old age and would get more money into the social care system," he said.
"I think that would be fairer because the big divide in our society is not between old and young, it is between those who have owned homes for 20 or 30 years and those who don't and potentially never will."
The Local Government Association said a debate on a long-term solution was "desperately needed" and urged political parties to put aside their differences.
"With people living longer and more people with disabilities needing support, increases in costs and decreases in funding, the current system of adult social care is at breaking point," said Ian Hudspeth, head of its Community Wellbeing Board
But The Kings Fund, a leading health tank, said the £2bn to £3bn likely to be raised through Mr Green's plan would simply not be enough to deal with the pressure on the system and more than twice that would be required to put it on a sound financial footing.