Trump insisted on Wednesday that the United States needed the vast Arctic island for national and international security, and has previously refused to rule out the use of force to get it.
"We have to have it," he said.
Russian President Vladimir Putin said Thursday he considers Trump's plans "serious".
"It is a deep mistake to think that this is some extravagant talk from the new American administration. It is nothing of the sort," Putin said.
Danish and Greenlandic officials, backed by the European Union, have insisted that the US will not get Greenland.
Danish Prime Minister Mette Frederiksen denounced plans by a US delegation to visit the Arctic island uninvited -- for what was initially a much broader visit to Greenlandic society -- as "unacceptable pressure" on Greenland and Denmark.
Greenlanders -- a majority of whom oppose US annexation, according to a January poll -- had also said they would give the delegation a frosty reception, with several protests planned.
In the end, Vance and his wife Usha will only visit the US-run Pituffik Space Base in the northwest of the island, accompanied by US Energy Secretary Chris Wright.
The delegation is to meet with US Space Force members and "check out what's going on with the security" of Greenland, Vance said in a video message.
The vice president angered Danes in early February when he said Denmark was "not doing its job (protecting Greenland), and it's not being a good ally".
A fuming Frederiksen quickly retorted that Denmark had long been a loyal US ally, fighting alongside the Americans "for many, many decades", including in Iraq and Afghanistan.
- Key missile defence cog -
The Pituffik base is an essential part of Washington's missile defence infrastructure, its location in the Arctic putting it on the shortest route for missiles fired from Russia at the United States.
Known as Thule Air Base until 2023, the base served as a warning post for possible attacks from the Soviet Union during the Cold War.
It is also a strategic location for air and submarine surveillance in the northern hemisphere, which Washington claims Denmark has neglected.
Vance is "right in that we didn't meet the American wishes for an increased presence, but we have taken steps towards meeting that wish", Marc Jacobsen, a senior lecturer at the Royal Danish Defence College, told AFP.
He said Washington needed to present more specific demands if it wanted a proper Danish response.
In January, Copenhagen said it would allocate almost $2 billion to beef up its presence in the Arctic and north Atlantic, acquiring specialised vessels and surveillance equipment.
Putin expressed concern Thursday that "NATO countries, in general, are increasingly designating the far north as a springboard for possible conflicts".
- 'Not for sale' -
Greenland is home to 57,000 people, most of them Inuits, and is believed to hold massive untapped mineral and oil reserves, though oil and uranium exploration are banned.
Energy Secretary Chris Wright, a former mining executive, told Fox News on Thursday he hoped the US and Greenland could cooperate on mining to "bring jobs and economic opportunity to Greenland and critical minerals and resources to the United States."
Trump's desire to take over the ice-covered territory, which is seeking independence from Denmark, has been categorically rejected by Greenlanders, their politicians and Danish officials.
While all of Greenland's political parties are in favour of independence, none of them support the idea of joining the US.
JD Vance's visit comes at a time of political flux in Greenland.
Following elections in March, the territory has had only a transitional government, with parties still in negotiations to form a new coalition government.
Initially, Vance's wife Usha was to attend a dogsled race in the town of Sisimiut, while various early reports suggested Wright and US national security adviser Mike Waltz would also take part in the visit.
"Our integrity and democracy must be respected without foreign interference," Greenland's outgoing Prime Minister Mute Egede said in a post on Facebook on Monday.
He recalled that the government had not "sent out any invitations for visits, private or official".
A visit to Greenland by Trump's son Donald Jr on January 7 had also been seen as a provocation.
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