The Office for Budget Responsibility, the independent body that gives ministers economic forecasts, calculated in March last year that increasing defence spending to 3% of GDP would cost by 2029-30 an additional £17.3bn per year.

Bee Boileau, research economist at the Institute for Fiscal Studies, estimated the extra needed would be less, about £13-14bn, once existing spending increases were taken into account.

Last year the UK spent about 2.3% of GDP on defence, around £66bn. The UK – along with all other Nato allies – is also committed to spending 3.5% of GDP on core defence by 2035.

In Munich on Saturday, Sir Keir made a sustained argument for greater defence spending to meet the threat from Russia.

“We must build our hard power because that is the currency of our age,” he said. “We must spend more, deliver more, and coordinate more.”

Greater defence spending, the prime minister argued, would mean less reliance on the United States: “We should deliver generational investments that move us from over-dependence to interdependence.”

It would also, he said, allow the UK to cooperate more with European allies to defend Ukraine.

“To meet the wider threat, it’s clear that we are going to have to spend more, faster,” Sir Keir said. “We’ve shown our collective intent in this regard as well, with the historic agreement to increase spending to 5% on security and defence. And we’re prepared to explore innovative solutions.”

One defence source said the prime minister’s speech read like an argument for increased defence spending with the announcement left out.

Whitehall sources said Morgan McSweeney, Sir Keir’s former chief of staff, was one adviser who had been pushing hard for extra defence spending. But since his resignation last weekend, Treasury concerns were said to have hardened. “The loss of McSweeney has changed the dynamic,” one official said.

Treasury sources said they did not recognise this account, saying there was no specific 3% plan that they were resisting. They said joint conversations taking place about future defence spending which, they insisted, were for the whole of government, led by the prime minister.

But other sources in Whitehall said the Treasury could still be tasked to come up with ways of finding the money.