In a phone call with Danish Foreign Minister Jeppe Kofod, Mr Pompeo “expressed appreciation for Denmark’s co-operation” as a US ally.
Mr Kofod welcomed the “frank, friendly and constructive talk”.
It came after Mr Trump said Denmark’s prime minister had been “nasty” when rebuffing his idea of buying Greenland.
Mette Frederiksen had called his ambitions of buying the autonomous Danish territory “absurd”. She said she was “disappointed and surprised” that he had cancelled his visit.
The trip was scheduled to take place on 2 September, at the invitation of Denmark’s Queen Margrethe II.
The US State Department said Mr Pompeo and Mr Kofod had a friendly phone call on Wednesday about the “postponement” of the president’s trip.
“The secretary expressed appreciation for Denmark’s co-operation as one of the United States’ allies and Denmark’s contributions to address shared global security priorities,” spokeswoman Morgan Ortagus said in a statement. “The secretary and Foreign Minister Kofod also discussed strengthening co-operation with the Kingdom of Denmark – including Greenland – in the Arctic.”
Mr Kofod said the talks had affirmed a strong bond between the two countries.
The US and Denmark “are close friends and allies with long history of active engagement across globe,” he wrote on Twitter.
How did we get here?
While praising Denmark as a “very special country”, Mr Trump said in a tweet on Tuesday that his planned visit would no longer go ahead because Ms Frederiksen had “no interest in discussing the purchase of Greenland”.
Mr Trump had earlier confirmed reports that he was interested in buying the autonomous Danish territory.
Speaking to reporters on the White House lawn on Wednesday afternoon, he took umbrage at Ms Frederiksen calling his plans “absurd”.
“I thought that the prime minister’s statement that it was absurd, that it was an absurd idea was nasty,” he said. “I thought it was an inappropriate statement. All she had to do is say no, we wouldn’t be interested.”
“She’s not talking to me. She’s talking to the United States of America. You don’t talk to the United States that way, at least under me,” he added.
Earlier on Wednesday, Ms Frederiksen had told reporters the idea of selling the resource-rich Arctic island had “clearly been rejected” by its leader, Kim Kielsen, “a position I share of course”.
How did Danes react to the cancelled visit?
The cancellation was described as a “farce” by the leader of the populist Danish People’s Party, Kristian Thulesen Dahl.
“What is this man thinking of though? And with grounds that are worthy of an April Fools’ joke,” he tweeted.
Danish Conservative MP Rasmus Jarlov accused Mr Trump of lacking respect for his country.
Former foreign minister Kristian Jensen said Mr Trump’s move had resulted in “total chaos”.
A spokeswoman for the leftist Red-Green Alliance, Pernille Skipper, said: “Trump lives on another planet.”
Pia Kjaersgaard, the populist former speaker of the Danish parliament, said it showed “rude behaviour to the Danish people and the Queen, who invited him.”
Why might Greenland appeal to the US?
Mr Trump has reportedly taken an interest in Greenland, in part, because of its resources, such as coal, zinc, copper and iron ore.
But while Greenland may be rich in minerals, it relies on Denmark for two-thirds of its budget revenue. It has high rates of suicide, alcoholism and unemployment.
The US has long seen the island, which sits along a direct route from Europe to North America, as being strategically important. It established the Thule air force and radar base there at the start of the Cold War, which now covers space surveillance and forms the northernmost part of the US ballistic missile early warning system.
Meanwhile, new Arctic sea routes are opening up as climate change is blamed for the accelerating thaw of ice in the region.
China has recently been taking an interest in the area, too.