But later Trump, who is due to join European leaders at the World Economic Forum in Switzerland, told a news conference on Tuesday all was well.
"We will work something out where NATO is going to be very happy and where we're going to be very happy."
Meanwhile, Britain and Denmark will hold defence talks in Copenhagen on Wednesday focusing on security in the Arctic region, the British government said, as Trump's designs on Greenland threaten to upend historic alliances.
Trump's ambition - spelled out in social media posts and mock-up AI images - to wrest sovereignty over Greenland from fellow NATO member Denmark has threatened to blow apart the alliance that has underpinned Western security for decades.
It has also threatened to reignite a trade war with Europe that rattled markets and companies last year.
In a Truth Social post on Tuesday after speaking to NATO Secretary General Mark Rutte, Trump said, "Greenland is imperative for National and World Security. There can be no going back - On that, everyone agrees!"
To drive home the message, he posted an AI image of himself in Greenland, holding a US flag. Another showed him speaking to leaders next to a map showing Canada and Greenland as part of the United States.
Asked later how far he was willing to go to acquire Greenland, Trump told reporters at the White House: "You'll find out."
But Trump said he had a lot of meetings scheduled on Greenland at the World Economic Forum annual meeting in Davos, Switzerland.
"I think things are going to work out pretty well," he said.
Separately, Trump leaked messages including from French President Emmanuel Macron, who questioned what Trump was "doing on Greenland". Trump had earlier threatened to hammer French wines and champagnes with a 200 per cent tariff.
The French president told the World Economic Forum the US designs on Greenland were "crazy".
Danish Prime Minister Mette Frederiksen said she would not yield to Trump's demands and abandon Greenland.
"The American president has unfortunately not ruled out the use of military force. And therefore the rest of us cannot rule it out either," she told reporters.
An agreement on sharing responsibility for the security of the Arctic and the North Atlantic could offer a way out of the stand-off, Lithuanian President Gitanas Nauseda told Reuters at the WEF in the Swiss ski resort of Davos.
European leaders took to the stage in Davos attempting to project the continent's strength, though it was not immediately clear how the European Union will eventually respond.
Macron said the EU should not bend to "the law of the strongest".
"We do believe that we need more growth, we need more stability in this world, but we do prefer respect to bullies," Macron told the meeting in Davos.
Some spoke of the importance of reducing European dependence for security on the United States. European Commission President Ursula von der Leyen described a "seismic change" that made it necessary to build a "new form of European independence".
Canadian Prime Minister Mark Carney said he strongly opposed any US tariffs linked to the Greenland issue.
The EU has threatened to hit back with trade measures. One option is a package of tariffs on 93 billion euros ($A162 billion) of US imports that could automatically kick in on February 6 after a six-month suspension.
Another option is the "Anti-Coercion Instrument" , known informally as the EU's "trade bazooka", which allows tough measures including restrictions on lucrative digital services provided by US tech giants in Europe. It has never yet been used, but Macron, who has raised the prospect of invoking it, insisted again on Tuesday it was on the table.
with DPA and AP
