DAVOS, Switzerland — Former President Donald Trump briefly floated the idea of the United States buying Greenland in 2019, a proposal that was widely mocked and ultimately abandoned. But the episode revealed a strategic interest in the island that has not gone away.

Now, even though Mr. Trump is no longer president, and the idea of a purchase is off the table, European officials are still grappling with the implications of a more assertive American presence in the Arctic. The region is becoming increasingly critically important as climate change opens up new shipping routes and access to natural resources.

“The Trump governance’s interest in Greenland was a wake-up call,” said a European diplomat who spoke on the condition of anonymity. “It showed us that the U.S. is willing to think outside the box when it comes to the Arctic, and that we need to be prepared for anything.”

The concerns are not just about a potential purchase. European officials are also worried about