Lord Lansley, health secretary from 2010-12, said he was "disappointed" that more funding was not announced in the Autumn Statement.
Tory council leader Izzi Seccombe said funding was needed now and councils had been "cut to the bone".
But Prime Minister Theresa May said ministers were already working to ease the situation.
Funding cuts to council-run social care in England have been blamed for a big increase in patients stuck in hospital beds because care cannot be arranged elsewhere
But the chancellor did not specifically offer new resources either for the NHS or social care when outlining the Treasury's plans on Wednesday, only confirming that ministers would be sticking with departmental spending announced last year.
Speaking in the debate following his statement, Mr Hammond referred to an extra £10bn in money for the NHS by 2020-21 - which itself has been questioned by the health select committee and others.
Lord Lansley, who was health secretary during the early years of the coalition government, told BBC Radio 4's World at One he had thought "a substantial change in departmental expenditure was pretty unlikely" in Wednesday's statement.
But he added: "Not being surprised doesn't mean I wasn't disappointed. The next two years are going to be incredibly difficult and I think the time is now for trying to put some measures in place to try and help health and social care through those next two years."
He said improving social care provision could help tackle many of the "underlying problems" of the NHS: "I don't think that necessarily demands a lot of extra public funds, what it does mean however is the implementation of the plan that we put together in the last Parliament from the Dilnot Commission."
The independent commission recommended that social care costs be capped at £35,000 so people did not face losing much of their assets - Lord Lansley said it was not implemented at the time "because the Treasury didn't like it".
The government allowed councils to increase council tax by 2% this year to spend on care - but councils have claimed this is not enough to plug the shortfall in their care budgets because of the wider squeeze on their funding.
Speaking on Thursday, Prime Minister Theresa May said: "I recognise the concerns and challenges around social care. By the end of this parliament, the local authorities will have £3.5bn more to spend on social care and of course we're putting more money into the health service as well.
"But we need to see the health service and social care working together that's why the health secretary and the local communities secretary are appraised of the need for the challenges faced and are looking at this issue."
NHS: Budget controlled by NHS England and pays for hospitals, GPs, district nursing and mental health care. Services are provided free at the point of need.
Care: Budget controlled by councils and pays for care home and nursing home places as well as support in the home for tasks such as washing and dressing. Services are means-tested so only the poorest get help towards their costs.
But Conservative Warwickshire County Council leader Izzi Seccombe told the BBC councils had been "cut to the bone" and the funds available did not meet the demands of paying a national living wage as well as the needs of an increasing elderly population.
"By the end of 2020 we will see that amount coming in but the need is now, it is a tapered amount which will increase to the end of 2020 - right now we have a £1.3bn gap which is not being funded."
And Siva Anandaciva, head of analysis at NHS Providers which represents NHS trusts, said funding was failing to keep pace with soaring demand for NHS services - illustrated by longer A&E and operation waiting times and fewer people getting council-funded social care. The lack of announcement from the chancellor had been "very disappointing".
Labour leader Jeremy Corbyn said he had been "utterly astonished" there had been no a mention of the of the NHS or of social care.
"There's an NHS funding crisis and there's a social care crisis in Britain. The solution has to be more money going in to it to pay for the services that are necessary and to reduce the level of bed blocking."