Organised by Denmark, the "Arctic Light 2025" exercise brings together more than 550 troops from NATO allies Denmark, France, Germany, Norway and Sweden.
"The current security situation requires us to significantly strengthen the armed forces' presence in the Arctic and North Atlantic," Danish Defence Minister Troels Lund Poulsen said in a statement.
Since returning to the White House in January, Trump has repeatedly said the United States needs the strategically located, resource-rich island for security reasons, which has sparked diplomatic tensions with Denmark.
The United States, which operates an airbase in the northern part of the territory, has sharply criticised Denmark for not sufficiently investing in Greenland's defence.
Lund Poulsen said the exercise, coordinated with the Greenlandic government, was "a good example of our common commitment to strengthening the armed forces' ability to address threats in the Arctic".
Later on Monday, he told broadcaster TV2 that Denmark was ready to invest a "double-digit billion amount" -- 10 billion kroner is roughly $1.5 billion -- to acquire the US-made maritime patrol and reconnaissance aircraft P-8.
The defence minister said ideally the acquisition would be made in co-operation with NATO allies.
"But if that is not possible, I am also willing to acquire the P-8 aircraft capacity ourselves," the minister said, noting it could be used to hunt submarines.
"We need to have a better picture of what is happening around Greenland and the Faroe Islands," he added, while noting that the "threat level is increasing".
Last week, Denmark also said it would invest some 58 billion kroner in European-made air and missile defence systems, citing lessons from Russia's war in Ukraine.
In late August, Danish broadcaster DR reported that at least three Americans linked to President Donald Trump had conducted influence operations across the Arctic territory.
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