Theresa May has told the head of the NHS that it will get no extra money despite rapidly escalating problems that led to warnings this week that hospitals are close to breaking point.

The prime minister dashed any hopes of a cash boost in next month’s autumn statement when she met Simon Stevens, the chief executive of NHS England, senior NHS sources have told the Guardian. Instead she told him last month that the NHS should urgently focus on making efficiencies to fill the £22bn hole in its finances and not publicly seek more than the “£10bn extra” that ministers insist they have already pledged to provide during this parliament.

She told him the NHS could learn from the painful cuts to the Home Office and Ministry of Defence budgets that she and Philip Hammond, the chancellor, had overseen when they were in charge of those departments, according to senior figures in the NHS who were given an account of the discussion.

Senior Whitehall sources have confirmed that Hammond’s statement on 23 November will contain no new money for the NHS, despite increasingly vocal pleas from key NHS organisations and the public’s expectation of extra health spending if Britain voted to leave the EU.

NHS Providers, which represents 238 NHS trusts, last week accused ministers of perpetuating “a bit of a fantasy world” on how well the NHS is doing after the worst-ever performance figures for key waiting time targets for A&E care, planned hospital operations and cancer treatments led to warnings that it was starting to buckle under the strain of unprecedented demand.

Health experts warned that the NHS would have to ration treatment, shut hospital units and cut staff if it gets no extra money soon.

Nigel Edwards, chief executive of the Nuffield Trust health thinktank, said: “If the government has firmly decided not to revisit NHS funding, this underlines that the health service faces four very difficult years. In particular, balancing the books in 2018 and 2019 when funding will flatline looks all but impossible with the current level of services.

“If more money from tax or borrowing is ruled out, the only choices left may be even less attractive, including reducing access and services, closures and reductions in staff,” he said.

Jeremy Hunt, the health secretary, and Jim Mackey, the chief executive of the health service’s financial regulator, NHS Improvement, also attended the 8 September meeting, which was Stevens’ and Mackey’s first encounter with the prime minister.

“No 10’s message at the meeting was quite blunt and stark, that there will be no more money. Theresa May and Philip Hammond say that they presided over big efficiency programmes at the Home Office and MoD and didn’t whinge about it. Their view is that the NHS is already doing very well, but that’s head in the sand stuff,” said one NHS insider who was among those briefed on the meeting.

NHS leaders privately fear that May’s remarks indicate that she will be much tougher on the service’s pleas for more cash than David Cameron and does not appear to appreciate the extent of its deepening problems. She is said to be sympathetic to the view of many senior Treasury officials that, as one NHS source put it, “always giving the NHS more money is throwing good money after bad, like pouring water on to sand”.

May’s stance raises questions over the future of Stevens, who is preparing to give evidence on the NHS’s finances to the Commons health select committee on Tuesday. The NHS boss, who had a close relationship with Cameron and George Osborne, has recently irritated No 10 by publicly questioning the accuracy of the government’s claim – which May repeated at prime minister’s questions on Wednesday – that the NHS will receive £10bn extra by 2020.

He told the public accounts committee last month: “The government would record it as £10bn. The health committee recorded it a little differently. There is an apples and pears issue there.”

Stevens has welcomed the fact that the £8bn boost Osborne pledged during last year’s general election campaign was “frontloaded” to give the NHS £3.8bn more this year, a rise of 1.7%, as he had requested. But he highlighted that the service had not got the sums it needed for 2017-18, 2018-19 and 2019-20. On current plans, it is due to receive increases of just 0.6%, 0.2% and 0.1% respectively, even though demand for core NHS services such as A&E care is rising at 3% or 4% a year.

Chris Ham, chief executive of The King’s Fund thinktank, said that any policy of providing no more money was unwise, “simply not credible” and would threaten standards of NHS care. “If these accounts are true, then it is clear that Downing Street does not yet fully understand the impact on patients of the huge pressures facing the NHS.

“The view from the top of government appears to be that the NHS has been given the extra money it asked for and should deliver what is expected of it. But this misses the point that demand for services is rising rapidly and the NHS is managing with the lowest funding increases in its history,” he said.

A Downing Street spokesman said he could not comment on what May, Stevens and Mackey had discussed because it had been a private meeting .