Andy Burnham has hailed a deal between Denmark and the US to “strengthen security in the Arctic” after the two countries resolved a months-long feud over Greenland.

The UK prime minister and his Danish counterpart Mette Frederiksen “welcomed the US’s commitment to the region” in a call on Sunday morning, Downing Street said.

Nato previously said the agreement would help stabilise the region, as Trump declared Denmark would grant the US “permanent control over security, and all other needs” in the semi-autonomous Danish territory in the Arctic.

Burnham’s comments come as he is expected to meet Trump for the first time at the UN General Assembly (UNGA) in New York this week.

The agreement comes after months of threats by Trump to “take” Greenland over national security concerns, citing its strategic location for defence purposes as well as its mineral wealth.

No text of the agreement has been released, but Trump said in a Truth Social post the deal came with “no cost to the United States” and meant it could “do what is necessary” to “secure and defend the security of Greenland”.

He said it included a rule that no US adversary could maintain a military presence in Greenland or “make sensitive investments in Greenland, without our express written approval”.

In a readout of Sunday’s phone call, a No 10 spokesperson said Burnham and Frederiksen “welcomed the US’s commitment to the region and underlined the strategic and shared importance of the High North for Euro-Atlantic security”.

The spokesperson added: “The Prime Minister congratulated Prime Minister Frederiksen and the Kingdom of Denmark on negotiating a deal benefiting the US, Greenland and Denmark, and said he would pass on his congratulations to President Trump.”

While Burnham praised the agreement, Liberal Democrat leader Ed Davey called it a “pretend” victory and a near carbon copy of the existing agreement.

“Donald Trump is trying to claim he’s got a success in Greenland, but actually, all he’s got is almost a carbon copy of the agreement that’s persisted since the 1950s,” he said at a Q&A with members at party conference.

“So let Donald Trump try to pretend he’s had a victory. The truth is, Europe, Nato, countries like Canada stood up to him and won.”

Trump had tried for months to purchase or annex the semi-autonomous territory, arguing that it was vulnerable to adversaries, including Russia and China.

In the wake of the deal, Frederiksen said she was “pleased that there is a prospect of a good agreement” for all three parties, which was expected to be signed at the UN General Assembly next week.

The deal “strengthens our common security in the Arctic and the North Atlantic area” while also recognising “the sovereignty and territorial integrity of the Kingdom and the right of the Greenlandic people to self-determination”, she said in a joint statement with Greenland’s Prime Minister Jens-Frederik Nielsen.

The agreement would still need to go through “the necessary parliamentary procedures to enter into force” once signed, the statement noted.

Burnham has spoken with Trump at least twice since entering Downing Street and No 10 said he intends to use the UNGA to “drive forward work on issues that matter to people at home” including the cost of living and the war in Ukraine.