It is the latest breakthrough on the peninsula ahead of a meeting between Kim and US President Donald Trump, who said Saturday that talks could take place within "three to four weeks."
A senior spokesman for South Korean President Moon Jae-in said Kim made the comments during a landmark summit Friday at the demilitarized zone between the two countries when Kim became the first North Korean leader to step into South Korean territory since fighting ended in the Korean War in 1953.
Kim told Moon during the summit that he had no intention of targeting the US or the South with nuclear weapons, the South Korean President's office said Sunday.
"The United States, though inherently hostile to North Korea, will get to know once our talk begins that I am not the kind of person who will use nuclear weapons against the South or the United States across the Pacific," Kim was quoted as saying by Moon's spokesman Yoon Young-chan on Sunday.
"There is no reason for us to possess nuclear weapons ... if mutual trust with the United States is built through frequent meetings from now on, and an end to the war and non-aggression are promised."
Friday's pageantry and face-to-face talks -- the first between leaders of the two Koreas since 2007 -- culminated in Kim and Moon issuing a joint statement committing to the "complete denuclearization" of the peninsula,
calling for the end of the Korean War, and heralding a "new era of peace."
Kim had previously announced the conclusion of North Korea's nuclear testing program and the shuttering of the Punggye-ri complex, saying on April 20 that it had already "completed its mission."
In the comments released by Moon's office on Sunday, Kim also refuted claims by Chinese scientists earlier in the week that parts of the site had been so badly damaged by previous explosions that it may now be unusable.
"Some claim we are closing down an unusable test site, but if they come and see, they will understand that there are two bigger tunnels than the existing test facilities and that they are in a very good condition," Kim said, according to Yoon.
President Moon "immediately welcomed" Kim's decision to make the process around the closure public, and the two leaders agreed to consult each other about the timeline for inviting South Korean and US experts and journalists "as soon as the North is ready," Yoon said Sunday.
Kim also announced during the summit that Pyongyang would change its time zone by half an hour to align it with Seoul time, reversing a decision made just three years ago in 2015
, Moon's office said Sunday.
"There were two different clocks in the reception hall at Peace House. One was for Seoul time and the other for Pyongyang time, which made my heart heavy," Moon's office quoted Kim as saying. "Let's first unify the two different times of the two Koreas."
'It may not work out'
Friday's Korean summit sparked a flurry of high-level phones calls between the United States and its Asian allies over the weekend.
During a phone call with Trump on Saturday, Moon "repeatedly" said the success of the summit was due to the strong support of his US counterpart, according to the South Korean government.
Moon and Trump also spoke separately to Japanese Prime Minister Shinzo Abe, who praised the success of the summit as long as it leads to "concrete action" on the part of North Korea.
Speaking at a rally in the US state of Michigan on Saturday night, Trump said it was difficult to predict how successful his own meeting with Kim might be.
"Look, I may go in. It may not work out. I leave," Trump said, adding that he couldn't tell the crowd what would happen "because we don't really know."
Although Kim has publicly said he is ready for the denuclearization of the Korean Peninsula, experts say his definition of the term may differ from the US and South's, and there is widespread skepticism that Kim would ultimately be willing to give up his nuclear weapons.
But US Secretary of State Mike Pompeo, who met Kim in Pyongyang for secret talks in early April, told ABC News that the North Korean leader was "prepared" to work towards denuclearization.
"I had a clear mission statement from President Trump. When I left Kim Jong Un understood the mission exactly as I described it today," Pompeo said in an interview with ABC News' Jonathan Karl that was scheduled to air on Sunday in the US.