The US president is said to have discussed the idea of purchasing Greenland, an autonomous Danish territory, during dinners and meetings with advisers.
In a statement on the island government’s website, a spokesman said: “We have a good cooperation with USA, and we see it as an expression of greater interest in investing in our country and the possibilities we offer.
“Of course, Greenland is not for sale.”
Greenland’s foreign ministry also dismissed the idea, saying: “We’re open for business, not for sale.”
A Trump ally told the Associated Press that the president had discussed the purchase but said they did not believe he was serious about it.
Greenland MP Aaja Chemnitz Larsen also dismissed the president’s reported interest.
“No thanks to Trump buying Greenland!”, she tweeted, adding that a “better and more equal partnership with Denmark” was the best way forward for the islanders.
Mr Trump’s reported plans were also dismissed by politicians in Denmark.
Former prime minister Lars Lokke Rasmussen tweeted: “It must be an April Fool’s Day joke...but totally out of [season]!”
Soren Espersen, the foreign affairs spokesman for the populist Danish People’s Party, said: “If he is truly contemplating this, then this is final proof that he has gone mad.
“The thought of Denmark selling 50,000 citizens to the United States is completely ridiculous.”
“Out of all things that are not going to happen, this is the most unlikely. Forget it,” Danish Conservative MP Rasmus Jarlov tweeted.
The story was first reported by the Wall Street Journal, which said Mr Trump had spoken about the purchase with “varying degrees of seriousness” – although other sources differed over whether the president was joking or seriously hoping to expand US territory.
The newspaper also reported that it was “unclear” how the US would go about acquiring Greenland if Mr Trump was not joking.
If serious, Mr Trump’s interest in Greenland was said to be in part because of its natural resources, such as coal, zinc, copper and iron ore.
More than 80 per cent of the island is currently covered by an ice cap, but the rising global temperature means this is fast receding, opening up new access to the mineral resources.
Though rich in minerals, Greenland currently relies on Denmark for two-thirds of its budget revenue, while its population of about 56,000 suffer high rates of suicide, alcoholism and unemployment.
The New York Times reported that people briefed on the discussions had said the president was also interested in Greenland’s “national security value” because of its location.
The US established a radar base there at the start of the Cold War, and the island has long been seen as strategically important
Republican Representative Mike Gallagher said Mr Trump’s idea was a “smart geopolitical move”, tweeting: “The United States has a compelling strategic interest in Greenland and this should absolutely be on the table.”
It would not be the first time an American leader tried to buy the world’s largest island.
In 1946, the US president Harry Truman proposed to pay Denmark $100 million to buy Greenland after flirting with the idea of swapping land in Alaska for strategic parts of the Arctic island.
Neither the White House nor Denmark commented on the report that appeared in the Wall Street Journal.