The US leader has raised the issue during meetings and dinners, discussing the advantages of owning the world’s largest island, an autonomous Danish territory, they said.
According to the aide, Mr Trump’s proposal for the US to snap up Greenland has each time come with “varying degrees of seriousness”.
Mr Trump is set to visit Denmark next month, but the prospect of such a purchase is not expected to be on the agenda.
Greenland is home to Thule Air Base, the US military's northernmost base, which is located about 750 miles above the Arctic Circle built in 1951.
The radar and listening post features a ballistic missile alert system that can warn of incoming intercontinental projectiles and reaches thousands of miles into Russian territory.
In 1946, the US 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 immediately commented on the proposals, which were first reported in the Wall Street Journal, when approached on Thursday.