The head of the NHS and the government are at loggerheads over how much the health service can be improved for the £20.5bn extra Theresa May has pledged to give it, the Guardian can reveal.

Simon Stevens, the chief executive of NHS England, has been having major disagreements behind the scenes in recent weeks with Downing Street, the Treasury and Department of Health and Social Care about how much the forthcoming NHS long-term plan can promise to boost care.

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“Tension” and “difficulties” have emerged during detailed horsetrading between the two sides amid sharp differences of opinion over the extent of the document’s ambitions, well-placed NHS and Whitehall sources have told the Guardian.

Negotiations have left ministers “fed up” and “deeply irritated” that Stevens is refusing to include explicit guarantees they believe will reassure voters that the service will improve dramatically over the next five years thanks to the extra money.

The plan, which will set out how the extra money will be spent, had been due to come out earlier this week but was delayed and is likely to finally appear in the week after next, subject to events at Westminster and further discussions between Stevens and ministers about its contents.

Ministers have told NHS England the plan should include specific annual improvements it will promise to make every year between 2019-20 and 2023-24 in its most challenging areas.

They want milestones written into it spelling out how close in percentage terms the NHS will get every year to once again meeting key waiting time targets covering A&E care, cancer treatment and planned operations, and also by how much the service’s dire finances will be turned round.

However, Stevens has left ministers frustrated by telling them privately that their ambitions are not realistic. Allies say he believes the £20.5bn more by 2023-24 is not enough for hospitals to get waiting times back on track after years of struggling to meet them and simultaneously honour headline-grabbing promises May and Philip Hammond have made recently, ahead of the plan being published, to expand and improve cancer and mental health care. They also want the money to pay for care to be transformed, with a major expansion of out-of-hospital services.

NHS England set up 14 different “workstreams” in the summer to draw up detailed proposals for how key areas of care needed to change to improve the nation’s health and keep the NHS sustainable, given the pressures of the ageing and growing population.

Stevens’s realism about the limits of the plan’s ambition has been reinforced by that process identifying improvements that would between them cost £80bn a year extra, four times the £20.5bn May has pledged. That has forced him to order a drastic culling of those proposals that are too costly to include in the plan.

Stevens has also warned them that the NHS’s chronic lack of staff – it is short of 103,000 doctors, nurses and other personnel – will also make it hard to drive the measurable progress they are seeking. Gaping holes in the NHS workforce are “dreadful and getting worse”, one senior figure said.

“Simon wants one thing and the politicians want another. The Treasury want to pin him to the floor over the action he will take to get all the waiting time targets back on track over the next few years, and he is resisting that. He wants flexibility,” said one source close to the discussions.

The Treasury is particularly exasperated by Stevens’s stance. But allies of the NHS chief say that he does not want to have his hands tied, sign up to timescales for progress that are likely to prove impossible to meet and to open himself up to criticism in the future for not delivering them.

One ally said: “The Treasury are the ones who are especially looking for high-profile and concrete improvements in care that the government can sell to the public in return for the £20bn. There is a lot of anxiety [among NHS leaders] because everyone knows the extra money is barely enough to maintain current standards, let alone transform services.”

Stevens is understood to feel unable to make public his reservations about how much progress ministers should expect for the £20.5bn given that he welcomed the money – which May gave to mark the NHS’s 70th birthday in July – at the time as “a change of gear, a step up” after eight years of tiny 1% annual increases. Its budget will rise from £115bn now to £135bn in April 2023.

Another NHS leader said: “Ministers want all the key targets back to where they used to be, the £1bn annual deficit down to zero and a host of new commitments delivered, all within the 3.4% annual budget rises over the next five years that the £20bn involves. But the numbers, and the whole thing, just don’t add up. You simply can’t get all those improvements on those timescales on 3.4%. It isn’t deliverable. But that’s what the government wants.”

NHS Improvement, the service’s financial regulator, is helping to draw up the plan. It warned last week that hospitals had already overspent by £1.23bn by the end of September, halfway through the service’s financial year, and that it may take five years to restore waiting time performance.

Niall Dickson, chief executive of the NHS Confederation, which represents organisations across the healthcare sector, urged ministers and the public to be realistic.

“The NHS long-term plan is a vital opportunity to improve patient care and change the way we deliver services to the public. But we should not underestimate how difficult it will be to recover performance on waiting times and to move NHS trusts and other organisations back into the black.

“We must be realistic about what is possible within the extra £20bn – the last thing we need is to set local services up to fail. And, above all, we will need a plan for securing the staff we need to respond to changing healthcare needs.”

NHS England denied a rift, saying: “The NHS, patient groups, clinicians and government are working closely together to finalise the NHS long term plan ready for publication before Christmas.” The Department of Health and Social Care also said there was no dispute, and they were “working closely with NHS England and NHS Improvement to develop an ambitious long term plan for our health service.”