Trump, who takes office on Jan. 20, also floated the idea of turning Canada into a US state, said he would demand far higher defense spending from NATO allies and promised to change the name of the Gulf of Mexico to the Gulf of America.
Still two weeks away from taking office, Trump has begun outlining an aggressive foreign policy with little regard to diplomatic considerations or the concerns of US allies.
Asked at a press conference at his Florida resort whether he could assure the world he would not use military or economic coercion as he tries to gain control of the Panama Canal and Greenland, Trump said, "No, I can't assure you on either of those two. But I can say this, we need them for economic security."
Trump criticised American spending on Canadian goods and military support for Canada, saying the US derives no benefits from doing so, and called the border between the two countries an "artificially drawn line."
He suggested he would impose tariffs on Denmark if it resists his offer to purchase Greenland, which he said is vital to US national security. Shortly before Trump's comments, his son Don Jr. arrived in Greenland for a private visit.
Denmark has said Greenland, a self-governing part of its kingdom, is not for sale.
"I don't think it's a good way forward to fight each other with financial means when we are close allies and partners," Denmark's Prime Minister Mette Frederiksen said late on Tuesday in response to Trump's comments.
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