Trump previously expressed interest in buying Greenland during his first presidential term and it seems he is still keen on the idea.
But the response from Greenland was blunt.
"Greenland is ours. We are not for sale and will never be for sale. We must not lose our long struggle for freedom," the island's Prime Minister Mute Egede said in a statement.
Trump on Sunday announced that he had picked Ken Howery, a former envoy to Sweden, as his ambassador to Copenhagen, and commented on the status of Greenland, a semi-autonomous part of Denmark and host to a large US Air Force base.
"For purposes of National Security and Freedom throughout the World, the United States of America feels that the ownership and control of Greenland is an absolute necessity," Trump wrote on Truth Social.
Trump, who takes office on January 20, did not elaborate on the statement.
The Danish government was not immediately available for comment.
Greenland, with its Pituffik air base, formerly known as Thule, is strategically important for the US military and its ballistic missile early-warning system since the shortest route from Europe to North America runs via the island.
Trump in 2019 expressed interest in buying Greenland, but the proposal was promptly rejected by Denmark as well as by the island's own authorities before any formal discussions could take place.
I promise not to do this to Greenland! — Donald J. Trump (@realDonaldTrump) pic.twitter.com/03DdyVU6HAAugust 20, 2019
Danish Prime Minister Mette Frederiksen at the time labelled Trump's offer as "absurd", leading him to term her dismissal of the idea as "nasty" and to subsequently cancel a visit to Copenhagen.
Since 2009 Greenland has held the right to declare independence from Denmark.
The island of some 56,000 inhabitants, which relies on significant budget transfers from Copenhagen each year, has so far refrained from doing so.
Separately on Sunday, Trump threatened to reassert US control over the Panama Canal, accusing Panama of charging excessive rates to use the Central American passage and drawing a sharp rebuke from Panamanian President Jose Raul Mulino.
