Trump is said to have asked his aides and the White House counsel to look into the possibility of buying Greenland, an autonomous Danish territory, according to media reports.
Reacting on the reports about the US President's plan, the Greenland government said in a statement: "Greenland is not for sale, and can't be sold, but Greenland is ready for negotiations to enter collaborations with other countries, including the US."
Similar comments were made by other politic in Greenland and Denmark.
"It must be an April Fool's Day joke...but totally out of (season)!", tweeted Denmark's former Prime Minister Lars Lokke Rasmussen.
Greenland is the largest island in the world (after Australia, which is defined as a continent in its own right). Eighty per cent of the island, located between the North Atlantic and Arctic oceans, is covered by an ice cap and has a population of about 56,000 people.
Greenland has a limited self-government and its own Parliament. It depends on the government in Copehnagen for foreign affairs and national security.
"It's a 'no thanks' from here", said Aaja Chemnitz Larsen, an MP from the the Inuit Ataqatigiit party, Greenland's second largest.
She said her homeland "was not a commodity that could just be sold" and that it was "tremendously uncomfortable to hear it discussed in such terms".
The Berlingske newspaper quoted Larsen as saying that it would be better all round for Greenland to stay with Denmark and eventually be granted full independence.
Asked if she did not find it flattering that Trump found Greenland attractive, she said: "It is Greenland's geostrategic location that Trump is interested in. Not the country itself, or Greenlanders. So no - it's not flattering."
Trump has reportedly taken an interest in Greenland, in part, because of its natural resources, such as coal, zinc, copper and iron ore.
"If he is truly contemplating this, then this is final proof, that he has gone mad," foreign affairs spokesman for the populist Danish People's Party, Soren Espersen, told national broadcaster DR.
"The thought of Denmark selling 50,000 citizens to the United States is completely ridiculous."
According to reports, Trump is planning to make his first formal visit to Denmark on September 2. He will meet the country's new Socialist Prime Minister, Mette Frederiksen - who herself will make her first official visit to Greenland next week.
Frederiksen has not commented on the reports.
The US has long seen the island as being strategically important and established a radar base there at the start of the Cold War.
US Republican Representative Mike Gallagher described Trump's reported idea as a "smart geopolitical move".
"The US has a compelling strategic interest in Greenland, and this should absolutely be on the table," he tweeted.
( With inputs from IANS )