Britain must “step up” and bolster its security, military and economic ties with Europe in the wake of President Trump’s threat to invade Greenland, the prime minister has said.

Sir Keir Starmer said that the world was in a “better place now than we were earlier this week” after the US president dropped his threat to unilaterally annex Greenland and impose tariffs on Britain and other European nations.

However, on Thursday Trump claimed to have secured “total access” to the semi-autonomous Danish territory for an unlimited period, after meeting Mark Rutte, the Nato secretary-general, on Wednesday.

“I’m not going to have to pay anything. We’re going to have total access to Greenland. We’re going to have all military access that we want,” Trump told Fox News on Thursday. “We’re going to be able to put what we need on Greenland because we want it.”

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Trump said the only expense was “the fact that we are building the Golden Dome”, a multibillion-dollar missile defence system that he envisages will be created before the end of his second term.

As details of the terms of the “framework” deal Trump said he had agreed with Rutte in Davos remained unclear, Denmark and Greenland said they had “red lines” which included sovereignty and democracy for the island.

Speaking on the way back to Washington to Davos, Trump was asked what period the draft framework might cover: “It’s infinity. The time limit is infinity, meaning there is no time limit. It’s for ever.” He added: “We can do anything we want. We can do military. We can do anything we want.”

Asked if the draft framework gave the US control of land in Greenland, Trump said the US “can do whatever we want. We’re all going to work together. And actually, Nato is going to be involved with us.”

Trump added that “everyone likes” the deal.

Jens-Frederik Nielsen, Greenland’s prime minister, said the territory “continues to be the centre of a serious situation. We are ready to discuss a lot of things and we are ready to negotiate a better partnership and so on, but sovereignty is a red line.”

A construction site with a crane in Nuuk, Greenland, below hillside homes and near the Sullorsuaq tunnel.

Nuuk, the capital of Greenland

JONATHAN NACKSTRAND/AFP/GETTY IMAGES

Stamer hosted Mette Frederiksen, the Danish prime minister, at Chequers on Thursday.

They said the security of the Arctic was “a matter for the entire Nato alliance” and Europe and Nato “would continue to drive forward progress”.

After the meeting Frederiksen left a message with a Beatles lyric in the guest book. “We’ll get by with a little help from our friends.” She travelled on to Brussels, where EU leaders were due to discuss Greenland.

Prime Minister Sir Keir Starmer and Prime Minister of Denmark Mette Frederiksen sit opposite each other in armchairs.

Frederiksen emphasised the sovereignty of Denmark over Greenland

STEFAN ROUSSEAU/PA

Frederiksen said that Nato should have a “permanent presence” in the Arctic, including around Greenland. “We said to the Americans a year ago that we can discuss our agreement on defence, but it has to be in the framework of us as a sovereign state,” she said.

Starmer said that threats from the US reinforced his view that in addition to keeping Nato together, Britain needed to “rise to the challenge” and build a closer relationship with the EU.

He told Channel 4 News: “It’s vital that we keep Nato together. It’s been the single most effective military alliance the world has ever known. And my focus this week has been on Nato unity and holding us together.

“But to rise to your challenge … I do think that Europe needs to do more, that Europe needs to step up. We need to be stronger on our own defence and security and that doesn’t just mean military capability and co-ordination.

“It goes to cyber, it goes to trade, it goes to energy. And it reinforces my argument that we, the UK, ought to be closer to Europe and we ought to have an urgent conversation about how we step up again in relation to defence, security, intelligence, collaboration and co-operation.”

How Nato angered Trump by accident — a timeline of Greenland tensions

The framework agreement on Greenland, reached by Rutte and Trump, contains five components, The Times has been told.

The 1951 Defense of Greenland Agreement, which already allows the US to build military installations, house troops and operate with almost complete freedom in certain areas, will be updated to guarantee American autonomy and immunity at its bases.

According to a diplomat briefed on the discussions, the other main points are that Nato will gain a command centre in Greenland to oversee an expanded role in the Arctic; a prosperity component will give the US the possibility of access to mineral resources; an exclusion agreement will rule out Chinese and Russian investment or other involvement on the island; and the US will have first refusal on acquiring Greenland if it becomes available. In return, the US agreed to drop the threat of tariffs and begin talks on the details.

Two children in pink winter gear pull a sled down a snowy street in Nuuk, Greenland.

Jens-Frederik Nielsen, Greenland’s prime minister, said sovereignty was a red line

SEAN GALLUP/GETTY IMAGES

Starmer said earlier that Trump’s shift in position was a “good thing” after weeks of pressure on the UK and allies in which Europe and America teetered on the brink of a trade war. He said it was a “reflection of pragmatism”, as Frederiksen praised the “British way of doing things”, saying: “You have a cup of tea and then you think a bit about everything.”

The prime minister also held a call with Rutte, who has been credited with brokering the deal that led Trump to drop his demands. Starmer told him the UK “stood ready to play its full part”.

Trump and Rutte were understood to have reached a verbal understanding about Greenland during their meeting at Davos. However, no document has yet been produced to describe a future deal, CNN reported, causing confusion among some Nato allies about the terms agreed.

The pair are said to have agreed to further discussions about updating the 1951 agreement. There was also speculation that parcels of land could be ceded to the US, but a Nato spokesperson said Rutte had not discussed this with Trump.

The president said the details were being negotiated but that “there’s no end, there’s no time limit” to US access, and he rejected any notion of leasing the territory from Denmark.

How do Greenlanders feel about Trump’s threats?

Leaders in Denmark and Greenland sounded an optimistic tone on Thursday after Trump’s apparent climbdown. He had also ruled out using military force against a fellow Nato member.

At a press conference in Nuuk, Greenland’s capital, Nielsen said he was not sure what was “concrete” in the deal, but expressed hope that a high-level working group established with officials from Greenland, Denmark and the US would resolve the differences.

“Respectful dialogue through the right channels is what we have been seeking from the get go, and I also feel that that’s the intention now from the other parties and I’m happy for that,” he said.

Speaking of Trump’s threats to acquire Greenland, he said: “The rhetoric has been unacceptable. It will affect how we think about each other going forward.”