Kim and Trump held a landmark summit in Singapore in June, in which they agreed to work toward complete denuclearization.
Chung Eui-yong (R), head of the presidential National Security Office, holds a press conference at the presidential office Cheong Wa Dae in Seoul on September 6, 2018 on the outcome of his one-day trip to North Korea the previous day as a special envoy of South Korean President Moon Jae-in. Negotiations with North Korea to get the isolated regime to give up its nuclear weapons have not been going well, but North Korean state media said Thursday that Kim remains committed to denuclearization.
North Korean leader Kim Jong Un (left) poses with South Korean President Moon Jae-in in April at the border village of Panmunjom.
However, Mr Kim has expressed a "sense of frustration" with the global community for not appreciating what he called Pyongyang's "very significant and meaningful" steps, Mr Chung said.
Kim emphasised that his "trust in Trump remains unchanged" despite the difficulties, Chung said, and expressed his intention to work closely with the USA to achieve denuclearisation "in the first official term of President Trump".
"Chairman Kim also emphasized he has never spoken ill of Trump to his aides or anyone really".
United States officials have insisted that a peace declaration, which many see as a precursor to the North eventually calling for the removal of all U.S. troops from the Korean Peninsula, can not come before North Korea takes more concrete action toward abandoning its nuclear weapons.
Kim also cited the dismantling of the Punggye-ri nuclear test site and Dongchang-ri missile launch facility as evidence of his commitment to end nuclear testing permanently, Chung said, The Washington Post reported.
They also discussed the complex process of dialogue between Pyongyang and Washington, Efe news reported.
Mr. Kim told Mr. Chung he still had faith in Mr. Trump despite diplomatic setbacks, and emphasised that he has not once talked negatively about Mr. Trump to anyone, including his closest advisors.
Last month, Trump called off Secretary of State Mike Pompeo's planned visit to North Korea - saying in a tweet that the trip was inappropriate at "this time, because I feel we are not making sufficient progress with respect to the denuclearization of the Korean Peninsula".
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The next big event between South and North Korea will take place over three days and two nights, . from September 18th through the 20th in Pyongyang.
But interpreting them required several caveats: Kim only said that he wanted to realize denuclearization before early 2021 - not commit to doing it - and he didn't provide a clearer definition for the term, something that has emerged as a sticking point with the U.S.
"If needed, we should pull forward the negotiations for the complete denuclearization of the Korean Peninsula with the development in relations between the South and North", Chung said. The two leaders will discuss measures on denuclearization, establishing peace and detailed steps to ease military tensions, Chung said.
The war ended with an armistice, not a peace treaty, meaning US -led United Nations forces, including South Korea, are technically still at war with the North.
The U.S. Embassy in Seoul on Thursday said it had no information to share on the matter.
US officials believed the Sony hack was retribution for The Interview, a comedy film that starred Seth Rogen and James Franco and centred on a plot to assassinate Kim.
The Korean War ended with an armistice agreement, which technically leaves the peninsula at war.
North Korean leader Kim Jong Un has vowed to keep working towards denuclearisation.
South Korea's liberal government is keen on keeping engagement with North Korea alive.
Trump spoke to Moon on the evening before Chung's trip and asked Moon to act as "chief negotiator" between Washington and Pyongyang, the spokesman said.