US President Donald Trump held open the possibility Wednesday for a resolution on his designs over Greenland after Denmark's top diplomat said he failed to change the administration's mind on wanting to conquer the island.

The foreign ministers of Denmark and Greenland, an autonomous territory under Copenhagen's sovereignty, met at the White House with Vice President JD Vance and Secretary of State Marco Rubio in a meeting the visitors had requested to clear up "misunderstandings" after Trump's bellicose language toward the fellow Nato member.

Trump, speaking after the meeting which he did not attend, for the first time sounded conciliatory on Greenland, acknowledging Denmark's interests even if he again said he was not ruling out any options.

"I have a very good relationship with Denmark, and we'll see how it all works out. I think something will work out," Trump said without explaining further.

He again said Denmark was powerless if Russia or China wanted to occupy Greenland, but added: "There's everything we can do."

Danish Foreign Minister Lars Lokke Rasmussen, speaking after leaving the White House, said a US takeover of Greenland was "absolutely not necessary."

"We didn't manage to change the American position. It's clear that the president has this wish of conquering over Greenland," Lokke told reporters.

"We therefore still have a fundamental disagreement, but we also agree to disagree."

He said the issue was "very emotional" for the people of Greenland and Denmark, a steadfast US ally whose troops died alongside Americans in Afghanistan and, controversially, Iraq.

"Ideas that would not respect territorial integrity of the Kingdom of Denmark and the right of self-determination of the Greenlandic people are, of course, totally unacceptable," Lokke said.

He nonetheless said the tone was "constructive" and said the sides would form a committee that would meet within weeks to see if there was possible headway.

Referring to the British prime minister who trumpeted his diplomacy with Hitler, Lokke said, "I am not a Chamberlain to say "Peace for our time," but we must seize the opportunities that present themselves."

While the talks were underway, the White House posted on X: "Which way, Greenland man?" The post included a drawing of two dogsleds – one heading towards the White House and a huge US flag, and the other towards Chinese and Russian flags over a lightning-bathed Kremlin and Great Wall of China.

Neither country has claimed Greenland, and Lokke said no Chinese ship had been spotted there in a decade and that there were no major Chinese investments.

Denmark promised ahead of the meeting to ramp up its military presence further in the vast, sparsely populated and strategically located island.

Trump has derided recent Danish efforts to increase security for Greenland as amounting to "two dogsleds."

Denmark says it has invested almost US$14 billion in Arctic security. Denmark also announced immediate military exercises that will include aircraft, vessels and soldiers, with Sweden also participating.

In another show of solidarity with Denmark following Trump's threats, Germany and France both said Wednesday they will send troops to Greenland. German's defense ministry said it would send a 13-person team. (AFP)