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"Denmark is a very special country with incredible people, but based on Prime Minister Mette Frederiksen's comments, that she would have no interest in discussing the purchase of Greenland, I will be postponing our meeting scheduled in two weeks for another time," Trump said on Twitter.
"The Prime Minister was able to save a great deal of expense and effort for both the United States and Denmark by being so direct," the president added.
Trump was due to visit Copenhagen in early September to meet with Frederiksen and Prime Minister Kim Kielsen of Greenland, which is an autonomous Danish territory.
"Greenland is not for sale. Greenland is not Danish. Greenland belongs to Greenland. I strongly hope that this is not meant seriously," Frederiksen told the newspaper Sermitsiaq during a visit to Greenland, Reuters reported.
Trump has on more than one occasion, and with "varying degrees of seriousness," floated the idea of buying Greenland, according to The Wall Street Journal. He is reportedly not the first American president to consider the prospect of purchasing Greenland.
For his part, Trump confirmed on Sunday that he is interested in buying Greenland, NBC News reported.
"It's something we talked about," Trump told reporters, according to NBC News. "Denmark essentially owns it, we're very good allies with Denmark." He referred to the theoretical transaction as "essentially" a "large real estate deal," and added that "strategically it's interesting and we'd be interested, but we'll talk to them a little bit."
NBC News reported Trump acknowledged that "first we have to find out whether or not they have any interest," and added that "they're losing a tremendous amount of money, so we'll see what happens."
Situated in the north Atlantic, Greenland is seen as an increasingly strategic location as melting ice opens up fresh shipping routes in the Arctic region. The land mass is also considered to be rich with coal, zinc, copper and iron ore.
Greenland's foreign ministry rejected the idea in a tweet, where it said that it is is "open for business, not for sale."
— Reuters and CNBC's David Reid contributed to this report.