Local authorities in England are to be allowed to bring forward council tax increases totalling 6% over the next two years to pay for social care.
Ministers have agreed the rise, which will be announced by Communities Secretary Sajid Javid on Thursday, adding a total of more than £90 to the average bill for a Band D property.
Former chancellor George Osborne announced last year that town halls would be permitted to add a 2% "social care precept" onto bills to raise £2 billion a year to help deal with the growing crisis in the sector.
Now authorities will be able to impose the total 6% rise over two years rather than three, meaning a 3% precept would be added to bills in 2017/18 and 2018/19 but 0% in 2019/20.
Chancellor Philip Hammond earlier this week denied reports that he had been blocked by Theresa May from announcing the move in last month's Autumn Statement because of concerns it would hit the "just about managing" families who the Prime Minister has said she wants to help.
Charities supporting elderly and disabled recipients of social care welcomed the change as a "step in the right direction", but said it would not fill a black hole in funding estimated to reach £2.6 billion by 2020.
The chief executive of Leonard Cheshire Disability, Neil Heslop, said: "We are pleased that the Government has acknowledged the severe strain that social care is under. Giving local councils more flexibility to raise revenue locally is a step in the right direction.
"However it is not an answer on its own and risks creating a postcode lottery. While extra money is always welcome, this will do little to plug the £2.6 billion social care funding gap by 2020, particularly as the areas with the greatest need for extra funding will raise the least through the precept."
Mark Atkinson, chief executive at disability charity Scope, said the crisis in social care amounted to a "national emergency", with more than half (55%) of the 400,000 working age disabled people who rely on care saying that they do not receive enough support to live independently.
"Disabled people have told us that they have resorted to sleeping in their wheelchairs, been unable to go to the toilet and have been forced to survive on one hot meal a day," said Mr Atkinson. "It's deeply worrying that disabled people are becoming more isolated, cut off from society, slipping into crisis and ending up in A and E.
"Following the omission of social care in the Autumn Statement, the reported 6% increase in council tax won't go far enough and isn't the long term solution needed to meet rising demand and costs."
John O'Connell, chief executive of the TaxPayers' Alliance, warned: "A 6% council tax rise will put unimaginable pressure on already struggling families without providing any real solution to the the social care crisis.
"Since 1997 Council Tax has gone up by at least 58% in real terms in England alone, so councils must remember that raising it even more will have a serious impact on household budgets."
A Department for Communities and Local Government spokesman declined to confirm or deny the 6% figure, describing reports of Thursday's announcement as "speculative".
Council tax on an average Band D property in England was £1,530 in 2016/17, and a 3% rise would increase it by £45.90 next year and the same amount again in 2018/19.
The social care precept comes on top of annual rises in council tax, which are capped at 2% unless the authority wins residents' support for a higher rise in a referendum.
Councils are not obliged to impose the precept, but a survey by the Local Government Chronicle last week found the majority intended to make full use of the 2% then available.
Areas where councils take full advantage of the permitted rises and precepts could now see total bills rise by 10% - or more than £150 - over the next two years, taking the average Band D levy to more than £1,680 a year.
Liberal Democrat health spokesman Norman Lamb said: "This is an inadequate solution from a Government that is lacking the ambition required to tackle the crisis in care we face.
"Letting councils put up taxes will raise some money in some areas - but we face a national crisis. This isn't about councils being able to supplement essential care, it is about providing it and that needs proper funding from central Government.
"It would be short-sighted - and stupid - to create a postcode lottery in care where wealthier councils can raise the money they need and poorer areas are left underfunded."
Labour leader Jeremy Corbyn went on the attack over social care funding at Prime Minister's Questions in the House of Commons, telling Theresa May to "get a grip".
Mr Corbyn said Mrs May should accept that there is "a crisis in social care" and scrap plans to cut corporation tax from 20% to 17% in order to plug the funding gap.
He accused ministers of "passing the buck on to local government" by using council tax rises to pay for social care in a way which would create a postcode lottery in funding.
"Raising council tax has a different outcome in different areas of the country," said Mr Corbyn.
"If you raise the council tax in Windsor and Maidenhead, you get quite a lot of money. If you raise the council tax precept in Liverpool or Newcastle, you get a lot less.
"Is she saying that older people, frail, elderly, vulnerable people are less valuable in our big cities than they are in wealthier parts of the country?"
He added: "This social care crisis forces people to give up work to care for loved ones ... makes people stay in hospital longer than they should and leads people into a horrible, isolated life when they should be cared for by all of us through a properly funded social care system. Get a grip and fund it properly, please."
The Prime Minister said she recognised that there were "immediate pressures on social care" which will be addressed by Mr Javid in the provisional local government finance settlement for England on Thursday.
But she added: "We also recognise that this is not just about money. It is about delivery. There is a difference in delivery across the country. We need to make sure that reform is taking place so we see best practice in terms of integration of health and social care across the country.
"But we also need to ensure that we have a longer-term solution to give people the reassurance for the future that there is a sustainable system that will ensure that they are receiving the social care that they need in old age. That is what the Government is working on."
Michelle Mitchell, chief executive of the MS Society, said: "We're pleased to see the Government has finally acknowledged pressures on social care funding. Shrinking social care budgets are leaving people with MS without the support they need to live independent and dignified lives.
"But this measure is simply not enough to meet the scale of the crisis. We're also worried that the precept will raise the least amount in the areas with the most need. The level of care that people receive shouldn't be a gamble based on where they live.
"Ultimately, it's Government, not local authorities, that has to take immediate responsibility for the long-term sustainability of the social care system."