The Danish foreign ministry initially said the meeting, on the sidelines of a NATO foreign ministers' gathering, was scheduled, before clarifying that was not the case.
"Work is in progress regarding a possible meeting between Lars Løkke Rasmussen and Marco Rubio in connection with the NATO Foreign Ministers' meeting in Brussels," the ministry said in a follow-up statement.
But despite strained relations between the two countries over the autonomous Danish territory of Greenland, the ministry specified that "neither Greenland nor the Arctic are planned to be discussed".
"It is expected that both the situation in Ukraine and the security of Europe will be on the agenda, topics which are also anticipated to be discussed at the NATO meeting itself," it said.
Trump claims the United States needs the vast Arctic island for its security and has refused to rule out the use of force to secure it.
"We'll get Greenland. Yeah, 100 percent," Trump told NBC News television on Sunday.
Greenland's new prime minister subsequently said the autonomous territory will decide its own future and will not become part of the United States.
"President Trump says the United States 'will get Greenland.' Let me be clear: The United States will not get Greenland. We don't belong to anyone else. We decide our own future," Jens-Frederik Nielsen said in a Facebook post.
While on a visit to a US military base in the Arctic territory on Friday, US Vice President JD Vance accused Denmark of not having "done a good job by the people of Greenland".
Løkke replied by saying Denmark not appreciate Vance's "tone".
"This is not how you speak to your close allies, and I still consider Denmark and the United States to be close allies," he said in a statement posted on social media.
"The 1951 defence agreement offers the United States many opportunities to have a much stronger military presence in Greenland. If that's what you want, let's discuss it," he also said.
Rubio will join two days of talks among NATO foreign ministers in Brussels, a preparation for a June leaders' summit in The Hague.
The new US administration has shown itself ideologically at odds not just with Denmark, but much of Europe. Vice President JD Vance made the Trump team's European debut in February by calling on Germany to stop shunning the far right.
Rubio will arrive hours after Trump is set to implement sweeping tariffs, part of an effort to remake the global economic order and shatter decades of efforts toward freer trade.
Most European allies are expected to respond quickly and strongly, leading to fears of a global trade war with an epicentre in a divided Western bloc.
Other than Canada, which Trump has mocked as the 51st US state, perhaps no ally has come under as much fire as Denmark.
Trump covets its Arctic territory Greenland, which is resource rich and strategically located.
Ahead of the summit in The Hague, Trump is pushing NATO members to show their commitment by raising defence expenditure to five percent of GDP -- more than any, including the United States, now spends.
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