The US president has suggested purchasing the vast Arctic territory on numerous occasions with staff after hearing about its natural resources and geopolitical importance, according to the Wall Street Journal.
He has even reportedly asked his White House counsel to explore the idea, although it remains unclear how negotiations would be carried out.
But residents in Greenland’s capital Nuuk met the proposal with short shrift.
“No thanks to Trump buying Greenland!” tweeted Aaja Chemnitz Larsen, a Greenlandic politician representing one of two seats for the island in the Danish parliament.
She urged the Danish authorities instead to create a “more equal partnership” with Greenland and to ultimately grant it independence.
One local simply responded, “Oh please God no”, at the prospect, while another tweeted, “We are not something you can just buy. Keep away from our country“, adding the hashtag “#Jerk”.
“If United States of America buy me and my Greenland fellows then what? Are you sure? @WhiteHouse @realDonaldTrump,“ tweeted one Nuuk expedition company employee.
Denmark’s foreign office told The Independent it had no comment “right now".
Mr Trump’s interest in the potential acquisition was reportedly piqued during a dinner with associates last spring, where one guest said Denmark was struggling to fund the £420m subsidy its sends the island each year, before suggesting the president should look to buy it.
“What do you guys think about that?” Mr Trump, who had shown “varying degrees of seriousness“ at the prospect, asked the room according to The Journal. “Do you think it would work?”
Some advisers are said to have supported the plan, believing it could have economic benefits and leave a legacy similar to that of the US’s acquisition of Alaska from Russia in 1867.
Others have played down the prospect, believing it to be nothing more than a fleeting interest of the president’s unlikely to come to fruition.
The US considers Greenland to be a militarily strategic location – its Thule Air Base on the northwest coast of the island works to detect potential intercontinental ballistic missile threats against North America.
Greenland is home to more than 55,000 people and has been Danish-owned since the early 18th century, before becoming a fully-integrated part of the country in 1953.
If he were to pursue the idea, Mr Trump would not be the first president to show interest in Greenland.
Harry Truman offered $100m to buy the island from Denmark in 1946, an offer that was rejected.