"We are open to criticisms, but let me be completely honest, we do not appreciate the tone in which it's being delivered," Foreign Minister Lars Lokke Rasmussen said in posts on social media.

"This is not how you speak to your close allies, and I still consider Denmark and the United States to be close allies," he said.

Vance made his comments during a trip to the Pituffik Space Base in northwestern Greenland, viewed by both Copenhagen and Nuuk as a provocation.

"Our message to Denmark is very simple: You have not done a good job by the people of Greenland," Vance told a press conference.

"You have under-invested in the people of Greenland and you have under-invested in the security architecture of this incredible, beautiful landmass," he added.

Trump argues that the United States needs the vast Arctic island for national and international security and has refused to rule out the use of force to secure it.

Asked about the potential use of force, Vance stressed the US administration did not think that "is ever going to be necessary".

"We think we're going to be able to cut a deal, Donald Trump-style, to ensure the security of this territory but also the United States of America," Vance said.

Denmark's Prime Minister Mette Frederiksen also hit back at Vance in a statement, saying his remarks about Denmark were "not accurate". She also said she will be visiting the island next week.

Vance was accompanied by his wife Usha, National Security Advisor Mike Waltz, Energy Secretary Chris Wright, Utah Senator Mike Lee and former Homeland Security Advisor Julia Nesheiwat, who is Waltz's wife.

- 'Unacceptable pressure' -

Danish and Greenlandic officials, backed by the European Union, have insisted that the United States will not obtain Greenland.

Frederiksen has condemned the US visit as "unacceptable pressure" on Greenland and Denmark.

Rasmussen said Denmark had stepped up investment in Arctic security, with Copenhagen announcing in January a $2 billion plan to boost its military presence in the region.

A majority of Greenlanders oppose US annexation, according to a January poll.

Saturday saw hundreds of people rally outside the US embassy in Copenhagen, some waving banners with slogans including "Trump/Putin stealing countries" and "Yankees go home" or "Make ice cubes not war."

"It's very important that we are as numerous as possible today," said one demonstrator, Kista Lynge Hoegh, a Greenland native.

"We are worried for our families who live in Greenland and we are concerned about what will happen to them if annexation occurs, as Trump has threatened."

"We want to come together to say the American rhetoric regarding Greenland and Denmark is unacceptable," said another Greenlander, Mimik Rosing.

The Pituffik base is an essential part of Washington's missile defence infrastructure, located in the Arctic on the shortest route for missiles fired from Russia at the United States.

Known as Thule Air Base until 2023, it served as a warning post for possible attacks from the Soviet Union during the Cold War.

It is also a strategic location for air and submarine surveillance.

Greenland is home to 57,000 people, most of them Inuits.

It is believed to hold massive untapped mineral and oil reserves, although oil and uranium exploration are banned.

- Unity call -

While all of Greenland's political parties are in favour of independence, none of them supports the idea of becoming part of the United States.

A new broad four-party coalition government was announced in Greenland just hours before the US delegation's arrival, following elections earlier this month.

"It is very important that we put aside our disagreements and differences because only in this way will we be able to cope with the heavy pressure we are exposed to from outside," said incoming prime minister Jens-Frederik Nielsen.

A statement from Danish premier Frederiksen's office said that during her April 2 to 4 visit she would greet the incoming government, with whom she promised to "pursue close and trustful cooperation".