Speaking to reporters in Copenhagen on Friday, Danish Foreign Minister Lars Lokke Rasmussen did not say who participated in the previous day's talks, according to the Ritzau news agency.
Lokke Rasmussen announced the formation of a working group to seek a solution to the Greenland question.
"We will not communicate when those meetings are because what is needed now is to take the drama out of this," the Danish minister said, emphasising the need for a "calm process".
Trump told the World Economic Forum in Davos on Wednesday that the US would not use force to acquire Greenland, after previously refusing to rule it out.
He also withdrew his threat of punitive tariffs against European countries that had opposed a US plan to purchase Greenland.
A framework for a future agreement on Greenland and the entire Arctic region had been established during a meeting with NATO Secretary General Mark Rutte, Trump said.
Danish Prime Minister Mette Frederiksen visited Greenland on Friday in a show of support for the Arctic island.
Walking off her plane at the airport in Greenland's capital Nuuk, Frederiksen was greeted by Greenlandic Prime Minister Jens-Frederik Nielsen.
The two were seen embracing on the tarmac before driving away.
Frederiksen flew straight to Nuuk from Brussels after meeting with NATO chief Mark Rutte earlier in the day to discuss how the military alliance can boost security in the wider Arctic region.
Greenland is an autonomous territory of the Danish kingdom. The two governments say the island is not for sale and its sovereignty is not up for discussion, but they are open to talks on other topics, such as security and economic development.
"We are preparing the next steps," Frederiksen told reporters while walking alongside Nielsen in central Nuuk. "I'm first and foremost here to show our strong support for Greenland's people as a difficult time."
Trump said on Thursday he had secured total and permanent U.S. access to Greenland after talks with Rutte, who said allies would have to step up efforts to ward off threats from Russia and China. The U.S. military is already permitted wide access to Greenland under treaties, although Washington has scaled back its presence there since the Cold War to a single small base.
The island's future has prompted a crisis in transatlantic relations after Trump demanded Greenland become U.S. territory, refused to rule out military force to obtain it and announced new tariffs on European countries that objected.
The crisis subsided after he lifted the threat to use force on Wednesday and withdrew the proposed tariffs on Thursday.
"We agree that NATO must increase its engagement in the Arctic. Defence and security in the Arctic is a matter for the entire alliance," Frederiksen said in a social media post, along with a photo of herself and Rutte in Brussels.
Rutte said he was working with the Danish leader to enhance deterrence and defence.
Denmark's foreign minister said on Friday that diplomats from Denmark and the United States had met in Washington on Thursday, establishing a plan for how to proceed.
"We will not communicate when those (future) meetings are, because what is needed now is to take the drama out of this... we need a calm process," Foreign Minister Lars Lokke Rasmussen said.
A source familiar with the matter said Rutte and Trump had agreed on further talks between the U.S., Denmark and Greenland on updating a 1951 agreement that governs U.S. military access and presence on the Arctic island. (Reporting by Jacob Gronholt-Pedersen and Janis Laizans in Nuuk, and Soren Jeppesen and Stine Jacobsen in Copenhagen, editing by Terje Solsvik and Toby Chopra)
with DPA
