The United States, Denmark and Greenland expect to sign an agreement on Arctic security at the United Nations General Assembly in New York next week, after US President Donald Trump announced on Friday, US time, that the deal gives Washington "permanent control over security, and all other needs, in Greenland".

The Danish Prime Minister's Office confirmed the agreement in a joint statement with Greenland's government, but described it in different terms. The three governments "are expected to sign an agreement on strengthened security in the Arctic and the North Atlantic Area next week", it said, after which it "will enter into force subject to the necessary national parliamentary procedures". The text has not been published.

What each side says the deal contains does not match. Trump's post, published at about 7.18am AEST on Saturday, said "no U.S. adversary can EVER have a base in Greenland, have a military presence in Greenland, or make sensitive investments in Greenland, without our express written approval", and that there would be "NO COST to the United States". He called it an "Infinite Life" agreement.

Danish Prime Minister Mette Frederiksen said the agreement "recognizes the sovereignty and territorial integrity of the Kingdom and the Greenlandic people's right to self-determination". Neither she nor Jens-Frederik Nielsen, chairman of Greenland's government, Naalakkersuisut, referred to US control, investment vetoes or new bases.

It is encouraging that we are about to make an agreement that will ensure and strengthen the security of Greenland, the Kingdom of Denmark, the United States, and the Western Alliance," Nielsen said. "The agreement recognizes Greenland's interests and our place in the international cooperation.

US officials added detail. A State Department official told Scripps News the arrangement would allow the US to "increase the number of bases in Greenland as required". Secretary of State Marco Rubio, in a statement carried by Scripps and AP, said the agreement "permanently guarantees U.S. security interests in the Arctic at zero cost to the U.S. taxpayer" and that Greenland "will forever be part of the strategic defense area of North America". Trump said the US would "immediately begin the process of developing a large Military presence" on the island.

The deal follows months in which Trump would not rule out taking Greenland by force. On 4 January, Frederiksen issued a statement saying the US had no right to annex any of the three countries of the Kingdom of Denmark, and pointing out that an existing defence agreement already gave the US wide access to Greenland. The US has operated what is now Pituffik Space Base in the island's north-west under a 1951 defence treaty, running missile warning and space surveillance for the US and NATO.

Until the text is released, the public record holds two accounts of the same agreement: a permanent US veto over who else can build or invest in Greenland, as Washington describes it, and a security arrangement that leaves Danish sovereignty untouched, as Copenhagen describes it.

Imran Bayoumi, of the Atlantic Council's Scowcroft Center for Strategy and Security, told AP the pressure campaign had cost the US in Europe. "I think Greenlanders, Danes, Europeans saw this as a real attack on their sovereignty, and it's going to take a lot of work to repair," he said.

No text has been released and no signing date has been set beyond next week's General Assembly high-level week. Once signed, the agreement must pass the necessary national parliamentary procedures before it takes effect.