Theresa May's claims of £10bn in extra funding for the NHS are untrue and give a "false impression" that the health service is awash with cash, MPs have warned.

In a strongly worded letter to the Chancellor, Dr Sarah Wollaston, chair of the Commons Health Select Committee, and four other MPs have urged Philip Hammond to use his first Autumn Statement as Chancellor to provide a cash injection for the NHS, amid claims social care in Britain approaching “tipping point”.

Dr Wollaston says the £10bn figure cited by Ms May is untrue at a time when pressure on its finances were threatening to become “overwhelming”. Ms May last used the figure at Prime Minister’s Questions after the Labour leader Jeremy Corbyn criticised the Government’s record on running the health service.

Dr Wollaston warned the Chancellor that unless the Government was able to put capital investment in the NHS on a sustainable footing, the service would struggle to meet the long-term demands of caring for an ageing population.

She said inflationary pressures on the NHS from a growing and ageing population meant that its needs were “far greater”.

She rejected suggestions the NHS could learn from the example of the cuts which Mr Hammond and Theresa May were able to drive through when they were at the Ministry of Defence and the Home Office.

“As the chief executive of NHS England [Simon Stevens] pointed out to us in oral evidence, despite the real-terms increases set out in the spending review, per capita funding for the NHS is projected to be flat in 2017-18, and to actually fall in 2018-19,” she wrote.

“That calls into question the ability of the NHS to maintain services in the latter part of the spending review period – and reinforces the danger of cutting spending on preventative and public health measures which could help slow the rate of demand growth.”

However, she said the most immediate problem was with adult social care, where the entire funding precept raised by local authorities through the council tax had been swallowed up meeting the demands of the living wages.

With the Care Quality Commission warning the sector was approaching a “tipping point”, she said that unless there was urgent action the knock-on effects of increased A&E attendances and longer hospitals stays could threaten the delivery of the NHS Five Year Forward View plan.

“Numerous sources testify to the impact of the real terms cuts to social care, not only to the vulnerable people who rely on care, but also on NHS service,” she wrote.

“There is an emerging consensus across the NHS that any additional money that might be available in the Autumn Statement should be directed first towards social care. We agree.”

Commenting on the letter, Jonathan Ashworth, the shadow Health Secretary, said: “The Tories have plunged our health and social care system into the worst funding crisis in its history. The responsibility for our social care system being at 'tipping point' lies squarely at their door.

“It's totally disingenuous and misleading for Jeremy hunt and Theresa May to claim they've given the NHS £10bn when it's manifestly untrue – and Dr Sarah Wollaston is right to take them to task for it.

“This Health Select Committee letter raises very serious questions about the sustainability of our health and social care system without extra funding. It's just not good enough for the Prime Minister to say 'there's no more money'. The Government need to sit up and listen to what everyone is saying and do what it takes to pull our NHS and social care system back from the brink. The test of the Autumn Statement will be whether it does so.”

A government spokesman said: “The Government has backed the NHS's own plan for the future with a £10bn real-terms increase in its funding a year by 2020-21, helping to ease the pressures on hospitals, GPs and mental health services. It is wrong to suggest otherwise.