Jeremy Hunt and Philip Hammond remain “gridlocked” over how much extra money the NHS should receive, hours before Theresa May is due to unveil a spending bonanza designed to rescue the ailing health service.

The health secretary and the chancellor have still not agreed the percentage increase to be received by the NHS in England from next April in a dramatic break with eight years of Conservative austerity policies. It is understood that May is keen to mark the 70th birthday of the health service on 5 July.

Hunt is holding out for a rise that would bring NHS England’s budget to between £157bn and £160bn – up to £32bn more than it gets now – by the end of this parliament in 2022-23.

But Hammond, worried about the potential impact of such a dramatic uplift in health spending on the wider public finances, is trying to limit the rise to a more affordable sum.

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Hammond, Hunt and Simon Stevens, the chief executive of NHS England, spent much of Friday in negotiations, conducted initially by telephone and in the afternoon at 11 Downing Street, the chancellor’s official residence. Their talks followed newspaper reports that May could soon hand the NHS a funding boost of anywhere between £4bn and £6bn a year.

The main sticking point between Hammond and Hunt is the percentage, somewhere between 3% and 4%, of the annual increase to be received by the NHS every year from 2019-20 until and including 2022-23, to honour the prime minister’s pledge of handing it a long-term funding deal.

“They are gridlocked between the completely different amounts [either 3.3% or 4%],” said a source close to the negotiations. A well-placed Whitehall source added: “They are not miles apart, but equally they are not inches apart either.”

Stevens, the Institute for Fiscal Studies, the Health Foundation and NHS bodies have made clear that the service needs a series of 4% rises, but a 3.3% increase is the very least it must receive. Hunt has endorsed the 3.3% figure as the bare minimum, while May also believes anything less will mean the NHS is unable to significantly improve its performance.

chart Source: Health Foundation

A 4% annual rise for the next four years would increase the NHS budget by £32.3bn to £160.5bn by 2022-23, an average of £8.1bn a year, according to analysis by the Health Foundation. But restricting the rise to 3.3% would bring the budget to £156.2bn, an annual increase of £7bn, leaving Hammond £4.3bn to spend in other areas, such as the military, police or schools.

Hunt and Stevens are hoping to persuade Hammond to agree a rise of at least 3.5%, and ideally above that, so the NHS can make major progress on improving cancer, mental health and maternity care and strides in other key priority areas.

The trio have to iron out the final details of the funding package before May announces the budget boost in an expected appearance on The Andrew Marr Show on BBC One on Sunday morning, and then gives more detail at an event with Hunt on Monday.

The risk for the government if Hammond does not agree to Hunt and Stevens’ demands has been underlined by new analysis by the Health Foundation, showing that limiting the rise to 3.3% would force patients to wait even longer for care.

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According to the foundation, that would lead to:

• The number of patients waiting for planned hospital treatment soaring from 4.2 million to a record 5.5 million.

• Even fewer patients requiring urgent cancer treatment receiving it within 62 days of being referred by their GP.

• And even larger numbers of A&E patients having to wait more than the supposed four-hour maximum to be treated and then admitted, transferred or discharged.

Prof Anita Charlesworth, the Health Foundation’s chief economist, said: “Annual increases of 3.3% a year above inflation are the minimum required to prevent services from further decline. But stemming decline should not be the goal for the NHS in its 70th year.

“After eight years of austerity, people are waiting too long for care, the NHS doesn’t have enough clinical staff, public health budgets have been cut and just four in 10 people with a mental health problem get any help at all.

“If we are to make any inroads into these problems and secure the health service for the future, funding needs to increase by at least 4% a year.”