Video report by ITV News Security Editor Rohit Kachroo
Greenland responded to reports Trump had spoken to aides about purchasing the territory from Denmark.
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.”
The story was originally reported by the Wall Street Journal, and it was met with mixed reactions on whether it was a joke or a serious proposal.
Greenland's Ministry of Foreign Affairs also made their point on Twitter, stating: "Greenland is rich in valuable resources such as minerals, the purest water and ice, fish stocks, seafood, renewable energy and is a new frontier for adventure tourism.
"We're open for business, not for sale."
Former Prime Minister Lars Løkke Rasmussen sent out a tongue-in-cheek tweet in response, stating: "It must be an April Fool’s Day joke ... but totally out of season!"
The territory's former foreign minister also spoke out against the news, tweeting: "Trump buying Greenland ?!
"Hopefully a joke, but otherwise terrible thought with the risk of militarization of Greenland and less independence for the Greenlandic people - besides great loss to Denmark."
The US president is due to visit Denmark in early September, of which Greenland is an autonomous territory.
A Trump ally revealed on Thursday that the president had discussed the possibility of buying the world's largest island, but they did not believe he was serious about it.
The White House has not responded to the report.
It wouldn't be the first time a country has tried to buy its way into owning part or all of Greenland.
In 1946, then-US president Harry Truman proposed to pay Denmark $100 million (£82.3 million) to buy Greenland, after flirting with the idea of swapping parts of Alaska for that in the Arctic.
China also recently bid for airport construction jobs on the territory, before Denmark intervened.
But could this mean more opportunity than offence?
Professor Ebbe Volquardsen from the University of Greenland agrees that the comments can be perceived as offensive to many Greenland locals.
"The idea to buy a territory and its people. That is rhetoric that belongs to imperialism 100 years ago, so of course it's disrespectful," said Professor Volquardsen.
However, he believes no matter how "disrespectful" Mr Trump's comments are, Greenland could use it as a bargaining chip for the future.
"It gives Greenlandic position - the position of such a small population - much more weight in negotiations with Denmark, with the USA in regards to trade and other projects.," he said.
"The Greenlandic government a should make use of this renewed interest in a smart and strategic way."