Trump tweeted a meme depicting the golden Trump International Hotel Las Vegas towering over modest, primary-colored houses along a rocky coast.
"I promise not to do this to Greenland!" he wrote.
Trump's son Eric also shared the meme on Instagram.
The meme first appeared on Twitter Thursday with the caption: "Greenland in 10 years."
It was shared after The Wall Street Journal reported the president's interest in buying the island.
On Sunday, Trump confirmed his interest in doing so but said it was not a priority for his administration.
"Strategically it's interesting and we'd be interested, but we'll talk to (Denmark) a little bit," he told reporters, adding that he viewed it as a "large real estate deal."
Denmark colonized the 772,000 square-mile (two-million square kilometer) island in the 18th century. It is home to only about 57,000 people, most of whom belong to the indigenous Inuit community.
Greenland's ministry of foreign affairs insisted Friday the resource-rich island was ready to talk business, but was not for sale.