Mr Trump confirmed to reporters on Sunday he had recently discussed the possibility, though he said such a move was not an immediate priority.
"The concept came up and ...strategically, it's interesting," Mr Trump told reporters from the tarmac in Morristown, New Jersey, as he prepared to board Air Force One.
"It is not No.1 on the burner; I can tell you that."
Mr Trump is due to visit Copenhagen early next month when the Arctic will be on the agenda in meetings with Frederiksen and Prime Minister Kim Kielsen of Greenland, an autonomous Danish territory.
White House economic adviser Larry Kudlow on Sunday initially confirmed media reports earlier in the week that Mr Trump had privately discussed with his advisers the idea of buying Greenland.
"I don't want to predict an outcome. I'm just saying the president, who knows a thing or two about buying real estate, wants to take a look at a Greenland purchase," Mr Kudlow told Fox News.
Mr Kudlow said the situation was "developing" and noted US President Harry Truman also had wanted to buy Greenland.
"And Denmark owns Greenland, Denmark is an ally, Greenland is a strategic place, up there. And they’ve got a lot of valuable minerals," Mr Kudlow added.
A defence treaty between Denmark and the United States dating back to 1951 gives the US military rights over the Thule Air Base in northern Greenland.
Greenland, located between the North Atlantic and Arctic oceans, is dependent on Danish economic support. It handles its own domestic affairs while Copenhagen looks after defence and foreign policy.
"It's an absurd discussion, and Kim Kielsen has, of course, made it clear that Greenland is not for sale. That's where the conversation ends," Frederiksen told the Danish broadcaster DR.
On Friday, Greenland foreign minister Ane Lone Bagger told Reuters: "We are open for business, but we're not for sale."