Trump announced the plan on social media moments before hosting a dinner for Republican governors at the White House on Saturday, where he sat next to and chatted with Landry.
"Working with the fantastic Governor of Louisiana, Jeff Landry, we are going to send a great hospital boat to Greenland to take care of the many people who are sick, and not being taken care of there. It's on the way!!!" Trump said.
Neither the White House nor Landry's office responded to queries about the post, whether the ship had been requested by Denmark or Greenland and which sick people needed help.
The Department of War had no immediate comment.
Danish King Frederik paid a second visit to Greenland in a year last week, an attempt to demonstrate unity with the territory in the face of Trump's push to buy the island.
Greenland, Denmark and the US late last month held talks to resolve the situation following months of tensions within the NATO defence alliance.
Trump's post came hours after Denmark's Joint Arctic Command said it had evacuated a crew member who required urgent medical treatment from a US submarine in Greenland's waters, seven nautical miles outside of Greenland's capital, Nuuk.
It was unclear what connection Landry had with the matter or if the post had any connection to the evacuation.
The US Navy has two hospital ships, the Mercy and the Comfort, but neither are stationed in Louisiana.
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