Trump says the strategically located and mineral-rich island is vital to US security and the United States must own it to prevent Russia or China occupying it.
He says all options are on the table for securing the territory.
"It is vital for the Golden Dome that we are building," he said in a social media post just hours before the meeting was due to start, referring to a proposed missile defence system.
Greenland and Denmark say the island is not for sale, threats of force are reckless and security concerns should be resolved among allies.
Prominent European Union countries have backed Denmark.
Trump also said that the NATO military alliance would become far more formidable and effective with Greenland in the hands of the US.
"Anything less than that is unacceptable," he wrote.
When Denmark's Foreign Minister Lars Lokke Rasmussen and his Greenlandic counterpart, Vivian Motzfeldt, meet Vance and Secretary of State Marco Rubio on Wednesday, their aim will be to de-escalate the crisis and find a diplomatic path to satisfy US demands for more control, analysts said.
"The end goal is to find some form of accommodation, or make a deal that would satisfy that need, or at least calm down the rhetoric sufficiently from Donald Trump," Andreas Osthagen, research director for Arctic and ocean politics at the Oslo-based Fridtjof Nansen Institute, told Reuters.
Noa Redington, an analyst and former political adviser to previous Danish prime minister Helle Thorning-Schmidt, said concerns were high in Denmark and Greenland that Motzfeldt and Rasmussen could be treated in the same way as Ukrainian leader Volodymyr Zelenskiy, when he suffered a public humiliation in a meeting with Trump - and Vance - at the White House in February 2025.
"This is the most important meeting in modern Greenland's history," he told Reuters.
Denmark and Greenland had originally sought a meeting with Rubio, hoping to have a discussion among top diplomats on resolving the crisis between the two NATO allies.
But Denmark's Rasmussen said Vance - who visited a US base in Greenland in March - had also wanted to participate and that the vice president would host the meeting himself, at the White House.
Greenlandic leaders appear to be shifting their approach in how they are handling the diplomatic crisis.
Until recently, they were stressing Greenland's path to independence.
But now their public statements put more emphasis on Greenland's unity with Denmark.
"It's not the time to gamble with our right to self-determination, when another country is talking about taking us over," Greenlandic Prime Minister Jens-Frederik Nielsen told Greenland daily Sermitsiaq in an interview published on Wednesday.
"That doesn't mean that we don't want something in the future. But here and now we are part of the kingdom, and we stand with the kingdom. That's crucial in this serious situation," he said.
Motzfeldt had a similar message.
"We choose the Greenland we know today - as part of the Kingdom of Denmark," she said in a statement released by the Danish ambassador to the US late on Tuesday.
