The Journal reports that Trump - a former real-estate developer - has had multiple discussions about purchasing Greenland, which is an autonomous territory of Denmark. CNN also has sources reporting similar information.
The President's interest has gone as far as to ask White House counsel if the purchase is possible, however, some believe it is only a joke.
Trump has been told the icy state has abundant resources and geopolitical importance according to the Journal's sources "familiar with the discussions". Obtaining the land would have substantial economic benefits, but has also been dismissed as unrealistic.
It may also provide Trump with a legacy similar to that of President Dwight Eisenhower who helped make Alaska a US state in 1959.
A real estate agent in Greenland told the Journal that a planned trip by Trump to Denmark in September has led to a running joke that he is coming to make an offer.
The suggestion has set Twitter alight, with many questioning how the purchase would work, how much Greenland would be worth and if the United States had enough money for the exchange.
"Has everyone forgotten that America is broke? Trump is leading the country into a $804 billion deficit. We'd have to borrow money from Denmark to buy Greenland from them," said one user. Another said: "Trump would want Greenland so he could claim he made the United States 25 percent bigger."
"No, Trump thinks Greenland is potentially a giant golf course."
Despite the initially stunned response of users, Trump's potential attempt to purchase the territory may not be the first by an American President. A Danish historian claims that Denmark's Government turned down an offer in 1946 from the United States.
Comment from the White House and Danish embassy in Washington has been sought by American media.