WASHINGTON—President Trump made his name on the world’s most famous island. Now he wants to buy the world’s biggest.

The idea of the U.S. purchasing Greenland has captured the former real-estate developer’s imagination, according to people familiar with the discussion, who said Mr. Trump has, with varying degrees of seriousness, repeatedly expressed interest in buying the ice-covered autonomous Danish territory between the North Atlantic and Arctic oceans.

In meetings, at dinners and in passing conversations, Mr. Trump has asked advisers whether the U.S. can acquire Greenland, listened with interest when they discuss its abundant resources and geopolitical importance and, according to two of the people, has asked his White House counsel to look into the idea.

Some of his advisers have supported the concept, saying it would be a good economic play, two of the people said, while others dismissed it as a fleeting fascination that will never come to fruition. It is also unclear how the U.S. would go about acquiring Greenland even if the effort were serious.

With a population of about 56,000, Greenland is a self-ruling part of the Kingdom of Denmark, and while its government decides on most domestic matters, foreign and security policy is handled by Copenhagen. Mr. Trump is scheduled to make his first visit to Denmark early next month, although the visit is unrelated, these people said.