Denmark has reportedly sent private messages to Donald Trump's team, letting the United States president-elect know it is willing to talk about letting Washington ramp up its military presence on the semi-autonomous island of Greenland, but that it will not cede control.

The communication followed Trump's repeated claims the US should control Greenland — a huge island with a population of fewer than 60,000 people — because of its strategic importance.

The Reuters news agency said the communication, which was first reported by the US news website Axios, aims to convince Trump the US can address any security concerns it has without forcibly occupying the massive, mineral-rich island.

Trump, who begins his term as president on Jan 20, has previously said he believes US control of Greenland is an "absolute necessity", and that he has not ruled out using military or economic force to do so.

Reuters said no one from Trump's transition team had commented on the Axios story, which quoted two unnamed sources who claimed they had seen the communication.

Denmark's Prime Minister Mette Frederiksen has already said she is willing to sit down with Trump to talk about his comments, likely after his inauguration.

Greenland's Prime Minister Mute Egede has also said he is prepared to talk to Trump, but that Greenland is only interested in full independence and not subjugation by another country.

Frederiksen and Egede met on Friday in the Danish capital Copenhagen to discuss the situation.

After the meeting, Egede told reporters: "Greenland is for the Greenlandic people. We do not want to be Danish, we do not want to be American. We want to be Greenlandic."

The latest twist followed Trump's son, Donald Trump Jr, visiting Greenland on Tuesday.

Pipaluk Lynge, a lawmaker from Greenland's largest political party who chairs its foreign and security policy committee, told the POLITICO news website Trump Jnr's visit did not include meetings with Greenland's government, and that encounters he had with locals should not be interpreted as indications of grassroots support for the US taking control of the island.

"No journalists were allowed to interview him. It was all staged to make it seem like we, the Greenlandic people … (would) love to be a part of the USA," she said. "People were curious, but some took pictures giving him (the) finger at the airport … Some wrote on Facebook: 'yankee go home'."

It is not the first time the US has coveted Greenland. In 1867, president Andrew Johnson mulled buying the island, as did president William Taft in 1910. And the US occupied Greenland during World War II, officially to prevent any other nation accessing it, a situation that ended after the war in 1946 with president Harry Truman again trying to buy it.