"We have asked NATO to be more present in the Arctic region," Frederiksen said at the start of a European Union summit in Brussels.

"Everybody in NATO agrees about that, the Arctic states, but also other member states, that we need a permanent presence from NATO in the Arctic region, including around Greenland."

Trump on Wednesday backed down from the threat of using force or tariffs to try to take over Greenland, after saying an agreement was reached in talks with NATO chief Mark Rutte.

Details remained scant of the accord -- but Trump did not make any progress towards his goal of trying to gain control over the autonomous Arctic territory of fellow NATO member Denmark.

Frederiksen said discussions about Denmark's sovereignty were off the table. "It cannot be changed," she said.

NATO said following the talks that the alliance would ramp up security in the Arctic, after Trump used the perceived threat from Russia and China to justify his desire for Greenland.

A source familiar with the discussions said Denmark and the United States would also look to renegotiate a 1951 defence pact on Greenland that governs American troop deployments on the island.

"We said to the Americans a year ago that we can discuss our agreement on defence, but it has to be in the framework of us as a sovereign state," Frederiksen said.

The Danish leader insisted the two sides "have to work together respectfully, without threatening each other".

"I, of course, hope to find a political solution within the framework of democracy and how we cooperate as allies," she said.