The health secretary has announced that the cap on NHS staff pay is to finally be scrapped – but has not said if the government will give the health service extra funding to cover the cost of whatever rise is finally agreed.

Jeremy Hunt told MPs on Tuesday that a seven-year run of 1% rises or pay freezes “wasn’t sustainable” and would come to an end with next year’s award for England’s 1.4 million NHS staff.

He said he had been given “leeway” to do so by the chancellor, Philip Hammond, but when questioned if the government would cover the cost he said: “That is something I can’t answer right now because the latitude that the chancellor has given me in terms of negotiating future pay rises is partly linked to productivity improvements that we will negotiate at the same time.”

That immediately raised concerns about whether the service will receive any extra funding increase in next month’s budget to help it foot the bill for a pay increase that has long been demanded by unions.

Simon Stevens, the chief executive of NHS England, made clear that the Treasury would have to boost the service’s budget as it could not afford to meet the pay rise given its tight finances.

“The secretary of state has said earlier today that the pay cap has been lifted,” he told the Commons cross-party health select committee. “That does need to be funded.”

Unions greeted Hunt’s move with deep scepticism. Dave Prentis, the general secretary of Unison, said scrapping the pay cap would only be meaningful if it led to a “proper” pay rise.

“The government’s announcement looks worryingly like a smoke and mirrors move, with talk of ‘productivity improvements’,” he said.

“NHS staff, patients and services shouldn’t be made to suffer to fund a pay rise.”

Fourteen health unions, including those representing nurses, midwives and healthcare assistants, recently said they wanted their members to receive a 3.9% pay rise in 2018-19 plus a further £800-a-head payment to help make up for the fall in the real value of their salaries since 2010.

They claim that NHS staff have seen their incomes eroded by as much as 15% since austerity began that year.

Kevin Brandstatter, a national officer with the GMB union, said: “If there is no new money then it’s a con trick to claim that the pay cap has ended.”

Jeremy Hunt has said he has been given leeway by the chancellor to end the cap. Photograph: Christopher Furlong/Getty Images

Hunt told the Commons that he hoped his announcement that the cap was being lifted would lead to a “win-win” for NHS staff and the government.

Hunt has helped persuade Theresa May and Hammond that the pay cap was a key reason why so many nurses, paramedics, doctors and other staff were leaving the NHS and why it was having such trouble recruiting enough replacements.

Stevens told the health committee that NHS staff deserved a pay rise but that the Treasury would have to cover the cost.

“Over time it will be necessary for NHS staff to get rates of pay that are consistent with the rest of the economy,” he said.

He also warned May and Hammond that the NHS will come under serious strain if the tiny budget increases it is due to receive in the next two years are not “amended” to help it meet rising demand.

“The budget position for funding currently pencilled in for the NHS for the next year and the year after looks extremely challenging and, if not amended, I think it is going to be very hard for the NHS to do all that has been asked for it over the course of the next year and the year beyond,” he said.

The NHS has £323bn less to spend than France and Germany, both of which put a greater proportion of their GDP into healthcare, he added.