Trump, who took office on Monday, set off alarm bells in early January by refusing to rule out military intervention to bring the Panama Canal and Greenland under US control.
"We are Greenlanders. We don't want to be Americans. We don't want to be Danish either. Greenland's future will be decided by Greenland," Prime Minister Múte Egede told a press conference, noting that Greenland faced a "difficult situation".
While Trump didn't mention Greenland in his inauguration speech on Monday, he was asked about it by reporters in the Oval Office afterwards.
"Greenland is a wonderful place, we need it for international security," Trump responded.
"I'm sure that Denmark will come along -- it's costing them a lot of money to maintain it, to keep it," he added.
Denmark's Foreign Minister Lars Løkke Rasmussen said earlier Tuesday that no country should be able to simply help themselves to another country.
READ ALSO: Danish government remains on alert after Trump inauguration
"Of course we can't have a world order where countries, if they're big enough, no matter what they're called, can just help themselves to what they want," Rasmussen told reporters.
On Monday, Danish Prime Minister Mette Frederiksen said in a post to Instagram that Europe would need to "navigate a new reality."
While noting the Greenlandic people's right to self-determination, the head of government also stressed the need for Denmark to maintain its alliance with the US -- which she described as Denmark's most important since World War II.
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