"Thank you, a very smart and gracious gesture!" Trump tweeted.
Washington is seeking the "complete, verifiable and irreversible denuclearisation" of the North and stresses that verification will be key.
Punggye-ri, in the northeast of the country, has hosted all six of the North's nuclear tests, the latest and by far the most powerful in September last year, which Pyongyang said was an H-bomb.
Kim has declared the development of the North's nuclear force complete and that it had no further need for the site.
All observation facilities and research institutes would be removed, along with guards, it said, "and the surrounding area of the test ground be completely closed".
Reporters from China, Russia, the United States, Britain and South Korea would be allowed to cover the event on site to show it "in a transparent manner".
Limits on foreign journalists were due to space constraints, it said, as the site was in an "uninhabited deep mountain area".
South Korea welcomed the announcement, which signalled the North's willingness to carry out its pledges "not just in words but in action".
"We hope the sound of the dynamite blowing up the tunnels at Punggye-ri will be the first salute in our journey towards a nuclear-free Korean peninsula," the South's presidential spokesman said.
Analysts said the move was positive but limited in its scope.
Given the stage it had already reached, Pyongyang "may feel like they don't need to test anything for a while", he said.
Jeffrey Lewis, of the Middlebury Institute for International Studies, expected that the North "will sanitise the site before letting anyone see it".
Tensions had been mounting for years as Pyongyang's nuclear and ballistic missile programmes saw it subjected to multiple rounds of increasingly strict sanctions by the UN Security Council, the US, EU, South Korea and others, and Trump last year threatened the North with "fire and fury".
But since the Pyeongchang Games, Pyongyang and Washington have agreed to the unprecedented Singapore meeting, and Kim has twice visited China to meet President Xi Jinping, after not paying respects in the six years since he inherited power from his father.
Kim and South Korean President Moon Jae-in last month affirmed their commitment to the goal of "realising, through complete denuclearisation, a nuclear-free Korean Peninsula" at a summit in the Demilitarized Zone that divides their countries.
But the phrase is a diplomatic euphemism open to interpretation on both sides.
Pyongyang has long wanted to see an end to the US military presence in -- and nuclear umbrella over -- the South, but it invaded its neighbour in 1950 and is the only one of the two Koreas to possess nuclear weapons.
This week the North released three Korean-Americans it had detained into the care of US Secretary of State Mike Pompeo, who was making his second trip to Pyongyang in two months.
However, Trump also withdrew the United States from a nuclear pact with Iran, raising questions over whether Pyongyang would trust Washington's promises.
Pompeo Friday promised the US would work to rebuild North Korea's sanctions-hit economy if it agreed to surrender its nuclear arsenal.
(This story has not been edited by Business Standard staff and is auto-generated from a syndicated feed.)