The island's government, however, said in a statement Friday it's not going to happen: "Greenland is not for sale."
Two of the people also told the Journal that Trump's aides were divided on the issue, with some praising it as solid economic strategy and others dismissing it as a passing fancy.
CNN has contacted the White House and the State Department for comment.
"They tried to buy us in 1867, during Second World War, and now they are trying again," local Kulusuk resident Bent Abeelsen told CNN. "Not gonna happen."
Aides expressed both expectation and reservation at the President's still-unclear interest in the idea and had questions about the island's military and research potential, the Journal reported. They pointed to those outside the administration floating a Greenland purchase as a potential legacy-builder for Trump, similar to President Dwight Eisenhower's statehood for Alaska, the paper added.
According to the Journal, one person described a dinner last spring where Trump told associates he had been advised to look into buying Greenland because Denmark faced financial trouble from supporting the territory.
"What do you guys think about that?" Trump asked the room, the person told the paper. "Do you think it would work?"
The person told the paper that Trump's comments seemed like more of a joke about his power than a genuine question. The person thought that the President was interested in Greenland due to its natural resources.
CORRECTION: This story has been updated to correct Eisenhower's role in Alaska's history.
CNN's Mary Ilyushina and Frederik Pleitgen contributed reporting from Greenland. Vasco Cotovio, Amanda Barnett and Jamie Ehrlich also contributed to this report.