It comes as Mr Kim and South Korea's president Moon Jae-in have agreed to a summit in North Korea from 18-20 September - the third between the two leaders this year.
Chung reported that Kim said he wishes for North Korea and the U.S. to put an end to their seven decades of hostile relations before the end of Trump's first term.
North Korea's Foreign Ministry on Tuesday published a lengthy statement on its website saying that an end-of-war declaration would be a necessary trust-building step between the wartime foes that would "manifest the political will to establish the lasting and stable peace regime on the Korean Peninsula".
The summit will discuss the implementation of a joint declaration reached by Moon and Kim when they met in the border village of Panmunjom in April, Director of National Security Chung Eui-yong, who led the team of envoys, said in a statement.
This will be the third inter-Korean summit this year.
Kim, however, expressed to Chung that he was frustrated by skepticism in the global community over his commitment to denuclearisation. Chung said there was no discussion with Kim about rescheduling that trip.
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After the summit, Trump declared that the North is "no longer a nuclear threat".
USA officials involved in the latest negotiations have said North Korea has refused to even start discussions about defining denuclearisation or other key terms such as "verifiable" and "irreversible", and has insisted the United States must first agree to simultaneous steps to reduce economic pressure.
Chung cites Kim as saying dismantling at the Tongchang-ri rocket site mean no future long-range missile tests would take place because it's his only long-range rocket testing site. "Chairman Kim especially stressed that he has never talked negatively about President Trump to his staff or anyone else".
"There is a still big gap between what the North considers sufficient goodwill gestures, like destroying its missile test stand or a nuclear test site, and what the USA wants, including on-site verification by experts", he said.
The U.S. Embassy in Seoul on Thursday said it had no information to share on the matter.
"But the work of convincing Chairman Kim to make this strategic shift that we've talked about for a brighter future for the people of North Korea continues", Pompeo said.
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News 1 reported Thursday the biggest US warship based in Asia could arrive in South Korea for joint exercises. North Korea has taken several steps such as dismantling its nuclear testing site this year, but the USA wants it to take more serious disarmament measures.
Seoul and Pyongyang both want a declaration to formally end the 1950-53 Korean War.
"It is the case that there is still an enormous amount of work to do".
Mr Kim was also pleased to receive a personal letter from Mr Moon, delivered by Mr Chung, KCNA added.
The cynical view of Kim's surprisingly upbeat comments to South Korean officials is that Kim is still making demands - he wants that Korean War peace treaty to boost his worldwide prestige and make it harder for the global community to maintain harsh sanctions against him-while offering little more than promises and flattery.
"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.
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