The two Tory leadership candidates have been challenged to bring forward plans to tackle the social care crisis in England if they become prime minister.
A committee of peers has called for an immediate £8bn cash increase and a move to a free, NHS-based system.
Jeremy Hunt has pledged more council funding and an opt-out insurance system for people to save for future care.
Boris Johnson said everyone deserved "security and dignity in their old age".
Meanwhile, councils say the government's delayed plans on elderly care should be published by September at the latest.
Ministers say they have responded to growing pressure on the system with extra money for residential care, providing £650m this year.
But the demand posed by an ageing society and the number of those living with debilitating and degenerative conditions, such as dementia, has put a huge strain on resources.
BBC assistant political editor Norman Smith said that, while both Tory leadership candidates had made the issue a priority, successive governments had ducked away from far-reaching changes, daunted by the costs and political risks involved.
The new plan, put forward by the Lords Economic Affairs Committee, was much more ambitious than any of the ideas mooted so far, he said.
The report focuses on social care in England only - the area is a devolved issue in Scotland, Wales and Northern Ireland.
It says residential care should be free at the point of delivery, modelled on the NHS. While individuals would still have to pay for their accommodation, there would be no costs for non-medical care.
The cross-party committee says the system should be paid for through general taxation, estimating the cost to be about £7bn a year.
Some of the cash, the peers say, could come from the £20bn increase in NHS funding by 2023 agreed by the government last year.
What is social care and how is it paid for?
Social care for the elderly covers non-medical needs such as help with washing, dressing and eating.
In England and Northern Ireland, everyone with income and assets worth 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.
In Wales the asset threshold is £50,000, having increased from the previous limit of £40,000 in April this year.
In Scotland, free nursing care is available to adults who have been assessed by the local authority as needing that service.
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.
Mr Hunt, who has described transforming care for the elderly as one of his "social missions", said a 10-year plan was needed to put the system on a sustainable financial footing.
He has said if he enters No 10 he will introduce a personal insurance scheme to allow people to save for care costs in old age, similar to the way pensions work.
He has said the system should be automatic, but people should be able to opt out.
Speaking on BBC Radio 4's Today programme, Mr Hunt said people should be "encouraged and incentivised" to save for their own care.
He added that greater integration of social care with the NHS was required so those with needs were not "pushed from pillar to post between one system and another".
"The other thing we need to is more money for local councils. I accept that there are funding difficulties", he added.
Mr Johnson said he had two priorities for social care: Ensuring no-one is evicted from their home to pay for care, and that everybody has security and dignity in their old age.
Asked if he could push forward with his plans without a Brexit deal agreed, he said: "We can get going on it right away if I am luckily enough to be elected.
"But this is something that will take time to work through. It is politically controversial… but we have got to do something."
A report by the Centre for Policy Studies think tank earlier this year called for wealthier homeowners to make a voluntary payment of up to £30,000 for their care needs in old age.
Speaking at a leadership hustings on Tuesday, Mr Johnson said politicians needed to come together, as a lasting solution was not possible without a cross-party consensus.
He said two principles should apply: that no-one should be at risk of losing their home, and that everyone should be treated with dignity in old age.
For Labour, shadow social care minister Barbara Keeley said the report was "another sign" that social care services were "at breaking point".
"The government's relentless and savage cuts to council budgets have caused a crisis with £7.7bn taken out of social care funding since 2010, cuts that even Jeremy Hunt now accepts were wrong and went too far. Boris Johnson must now do so too," she added.
The Local Government Association, which represents councils in England and Wales, urged the government to publish its adult social care green paper by September.
Its publication has been delayed several times.
Ian Hudspeth, chairman of the LGA's community wellbeing board, said: "More than 12 months has passed since the government announced yet another delay to the publication of its social care green paper.
"Those who rely on vital care and support cannot wait any longer."