Senior officials of Denmark and Greenland who flew to Washington to meet their US counterpart on Wednesday have failed to dissuade the Trump administration to drop its demands to own the world's largest island "one way or another", including militarily means.

Denmark's Foreign Minister Lars Lokke Rasmussen said the tone of the meeting was frank and constructive but conceded a "fundamental disagreement" between the two. "We didn't manage to change the American position," he told reporters in Washington. "It's clear that the president has this wish of conquering over Greenland."

He rejected Trump's claims that Greenland will have "Chinese destroyers and submarines all over the place" if the US doesn't control the territory and suggested the face-to-face talks had taken down the temperature.

"We had the opportunity to challenge the narrative of the president," Euronews quoted him as saying, adding the "narrative that we have Chinese warships all around the place" is not true. "There hasn't been one (Chinese ship) for a decade or so," he said citing Danish intelligence.

Rasmussen also announced the formation of a "high-level" working group with the United States government to find a "common way forward" with regard to US territorial designs on the Danish Arctic territory.

Rasmussen and his Greenlandic counterpart Vivian Motzfeldt held talks with US Vice-President JD Vance and Secretary of State Marco Rubio at the White House on Wednesday, following comments from President Donald Trump about his desire to take control of the strategically important vast frozen island, which has a population of just under 60,000 people, but huge mineral resources.

Trump has long said Greenland should be under US control, and has become increasingly strident in his demands, despite them putting his country at loggerheads with Denmark, a long-standing ally and fellow member of the NATO military alliance.

"NATO becomes far more formidable and effective with Greenland in the hands of the United States," Trump wrote on social media. "Anything less than that is unacceptable."

However, senior Democratic US senator Chris Murphy among others believed that the threat to annex Greenland represents an existential crisis for Nato.

Rasmussen called Wednesday's talks "frank but constructive", adding "we still have fundamental disagreements" and that he had failed to get the US to change its position, but he was keen for dialogue to continue.

He added that he knew Trump from his former role as prime minister of Denmark, and that the prospect of the US acquiring Greenland was "absolutely not necessary".

Speaking for Greenland, Foreign Minister Motzfeldt said it was vital to "normalize" international relations, and added that Denmark and Greenland, who have worked with the US for "many, many years," are keen to continue to do so.

Before the talks, Greenland's Prime Minister Jens-Frederik Nielsen reiterated that his country has no interest in being controlled by the US.

"One thing must be clear to everyone," he said. "Greenland does not want to be owned by the United States. Greenland does not want to be governed by the United States. Greenland does not want to be part of the United States."

There have been reports of financial incentives being offered to individual citizens, or even the use of military force to try and gain US control of Greenland, but a recent opinion poll by The Associated Press showed that just 6 percent of residents wanted it to become part of the US.

Pointedly, on the same day the talks took place, Denmark significantly increased its military presence in Greenland. These troops will soon be joined by reinforcements from Sweden, with France and possibly Germany also expected to help, while Norwegian government sources have spoken of "(mapping) further cooperation between the allies".

Motzfeldt says that it was a "core priority" that the defense and security of Greenland "are strengthened, that this is achieved in close cooperation with our NATO allies."