The agreement, to be signed at the UN General Assembly in New York at 10:30am on Tuesday (12:30am AEST Wednesday), has not been made public.
Denmark and Greenland say the deal will benefit the Arctic island and respect its sovereignty.
The deal is expected to allow the US, which already has one military base in Greenland, to expand its military presence to two additional sites.
One of those sites is Narsarsuaq, whose population has dwindled to a few dozen people since the US closed its Cold War-era Bluie West One military base there in the 1950s.
The other is Mestersvig, a military outpost currently used by the Sirius Dog Sled Patrol, a special forces unit that enforces Danish sovereignty in the territory, said two people familiar with the matter.
Trump will meet with Danish Prime Minister Mette Frederiksen and Greenland Prime Minister Jens-Frederik Nielsen in New York for a trilateral signing ceremony, a White House official said.
The deal follows months of pressure from Trump, who has said the US should control Greenland and at times refused to rule out using military or economic force to acquire the semi-autonomous Danish territory.
His threats peaked in January, triggering a diplomatic crisis within the North Atlantic Treaty Organisation, the military defence alliance, and pushing Denmark and Greenland into negotiations aimed at defusing the pressure.
The agreement "will be in a form where responsibility for security in the Arctic is not only Danish or American," Danish Foreign Minister Lars Lokke Rasmussen told national broadcaster TV2 on Sunday.
He said that to a large extent "it would be NATO that looks after the Arctic."
It remained unclear whether the deal amends an existing 1951 defence agreement that is the legal basis for the US presence in Greenland.
"It appears that we are now using this crisis to try to strengthen NATO's cohesion through Greenland," said Ulrik Pram Gad, a researcher at the Danish Institute for International Studies (DIIS).
Washington's strategic interest in Greenland dates back to World War Two.
In 1941, the US signed a deal with Danish envoy Henrik Kauffmann, who acted without authority from his country, then under German occupation.
The US sought the deal to keep Greenland out of German hands and secure the North Atlantic supply route between North America and Europe.
Under the agreement, the US built airfields, naval facilities, and weather stations across the island, reaching around 5500 military personnel by 1943, according to DIIS.
In 1946, President Harry Truman sought to buy the island as a strategic asset for $US100 million ($A140 million) in gold, but Copenhagen declined to sell.
Instead, a new agreement was signed in 1951 that gave the US military extensive rights in Greenland but was tied to the existence of NATO.
"I don't think Denmark and Greenland will sign if the same NATO clause isn't included," said Gad.
During the Cold War, the US maintained 17 military installations and more than 10,000 personnel in Greenland. Today only Pituffik Space Base remains, with some 150 US military personnel.
A NATO spokesperson said on Saturday that the alliance welcomed the new agreement, which would "help advance security, stability and co-operation in this vital region".
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