Theresa May’s claims that the government is putting £10bn extra into the NHS are untrue and the underfunding of the health service is so severe that it may soon trigger rationing of treatment and hospital unit closures, a group of influential MPs have warned Philip Hammond.

Five MPs led by the Conservative Dr Sarah Wollaston, the chair of the Commons health select committee, have written to the chancellor demanding the government abandon its “incorrect” claims of putting £10bn into the NHS annual budget by the end of this parliament and admit the severity of its financial shortage.

“The continued use of the figure of £10bn for the additional health spending up to 2020-21 is not only incorrect but risks giving a false impression that the NHS is awash with cash,” Wollaston and four fellow committee members tell the chancellor in a letter.

“This figure is often combined with a claim that the government ‘has given the NHS what it asked for’. Again, this claim does not stand up to scrutiny as NHS England spending cannot be seen in isolation from other areas of health spending.”

The letter’s other signatures are Dr James Davies, a Conservative MP who is also a family doctor; Labour’s Ben Bradshaw, a former health minister, Labour MP Emma Reynolds; and Dr Philippa Whitford of the Scottish National party, who is an NHS breast cancer specialist.

Their letter’s detailed rejection of the government’s claims raises serious questions about the accuracy of May’s insistence, in a newspaper interview on 17 October and again at prime minister’s questions two days later, that her administration was giving NHS England boss Simon Stevens even more than he had sought in negotiations with ministers.

May told the Manchester Evening News: “Simon Stevens was asked to come forward with a five-year plan for the NHS. He said that it needed £8bn extra; the government has not just given him £8bn extra, we’ve given him £10bn extra. As I say, we have given the NHS more than the extra money they said they wanted for their five-year plan.”

However, the MPs say that May’s £10bn claim cannot be justified. “The £10bn figure can only be reached by adding an extra year to the spending review period, changing the date from which the real terms increase is calculated and disregarding the total health budget,” they concluded.

In the run-up to the general election, George Osborne, the then chancellor, promised to spend £8bn more a year by 2020, a figure that has risen since. But the MPs dispute that arithmetic, saying that the real amount of extra cash being given to the NHS in England between 2014-15 and 2020-21 is only £6bn and even that much smaller sum has only come from cutting spending on public health programmes and medical education and training by £3.5bn.

Worries about health service funding have emerged with increasing intensity in the run-up to the autumn statement on 23 November after it emerged that May told the head of the NHS in private that it would get no additional money this parliament.

Last year, finances were so tight that the NHS overspent its budget but public pressure to fund the health service generously remains strong. During the EU referendum campaign, the successful leave campaign promised to boost funding for the health service by diverting money that it said was being spent in Europe.

Warning of the political risk involved in underfunding the NHS, the five MPs add that “public expectations of the health service, and the continued rise in demand for its care produced by an increasing and ageing population, mean that measures which could be taken in some government departments are not acceptable in the NHS … including rationing of care and cuts in service provision.”

The MPs maintain that what they see as short-sighted cuts to social care threaten the viability of NHS services. They also raised the risks of the Department of Health “repeatedly raiding” the NHS’s capital budget in recent years and the decision to give the NHS only tiny budget increases in 2017-18 and in the two years afterwards.

“Our fear is that, given the ‘U-shaped’ trajectory of increases in funding for the NHS over the spending review period, these short-term pressures will become overwhelming. 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 actually to fall in 2018-19. That calls into question the ability of the NHS to maintain services in the latter part of the spending review period,” they say.

Andrew Lansley, the health secretary in the coalition government, recently called for the NHS to be given £5bn more than the money already planned.

There have also been widespread calls for the government to make good on the suggestion by Brexit campaigners that leaving the EU could add £350m-a-week to the NHS budget.

NHS England declined to comment on the letter.

Chris Hopson, the chief executive of NHS Providers, which represents hospitals, said that NHS underfunding meant that “it is being asked to deliver an impossible task. Put simply, the gap between what the NHS is being asked to deliver and the funding it has available is too big and is growing rapidly”, he said.

Prof John Appleby, the chief economist at the Nuffield Trust health thinktank, said the MPs were right to warn that cutting the amount of per capita funding for healthcare could mean major restrictions to NHS services being needed in the later years of this parliament, too.

“It is hard to see how this can be reconciled with providing high quality healthcare that meets the needs of a growing and ageing population,” Appleby said. “Something will have to give – whether that’s an explosion in waiting lists, patients not being able to access new drugs coming on-stream or another record set of hospital deficits.”

The government rejected the MPs’ analysis and repeated previous statements made by May and the health secretary, Jeremy Hunt, including the highly contentious £10bn claim. “The government has backed the NHS’s own plan for the future with a £10bn real terms increase in its annual funding by 2020-21, helping to ease the pressure on hospitals, GPs and mental health services. It is wrong to suggest otherwise”, said a government spokesman.

“As the chief executive of NHS England said last year, the case for the NHS has been heard and actively supported. We have allowed local government to increase social care spending in the years to 2020, with access to up to £3.5bn of new support by then.”

A report released today by the Public Administration and Constitutional Affairs Committee, found that a lack of integration, caused by the historic split between health and care, meant that interdependent services were being managed and funded separately. This “political maladministration” was causing suffering for patients and relatives, it said.

The committee was responding to work carried out by the Parliamentary and Health Service Ombudsman, which it said highlighted “harrowing cases that illustrated the human cost of poor discharge”. These cases were not isolated but were persistent problems across the health service.

Poor patient discharge can take the form of delayed transfers of care, where patients are kept in hospital longer than is necessary, and premature or early discharge, where patients are sent home before it is clinically safe to do so, or without appropriate support in place.

Barriers to the implementation of best practice are prevalent at the interface between health and social care, the committee said. Pressures on resourcing and capacity were “leading to unsafe discharge practices”, and it called on health and social care leaders to ensure that person-centred care remained the undisputed priority.

The report found that while excellent guidance was available, good practice was not being applied equally across the system and more data was needed on the scale and impact of the failures.

Responding to the report, parliamentary and health service ombudsman Julie Mellor, highlighted the human cost that could arise when people fell through the cracks, and blamed the underfunding of social care.

She said: “We see too many cases where discharge from hospital has gone horribly wrong, particularly for older, frail people who often don’t have the right support in place at home to cope on their own.

“These shocking failures will continue to happen unless the government tackles the heart of the problem – the chronic underfunding of social care which is pilling excruciating pressure on the NHS, leaving vulnerable patients without a lifeline.”

Committee chair Bernard Jenkin said some hospital staff felt under pressure to discharge people earlier than was appropriate.

“Hospital leadership must reassure their staff that organisational pressures never take priority over person-centred care,” he said.

He stressed that staff needed to feel a level of trust and openness that enabled them to raise concerns about unsafe discharge.

The report referred to the Better Care Fund and the Discharge Programme Board as being “promising”, in bridging the gap between health and social care. But, it cautioned that these plans were far from implemented.