The United States continues to actively prepare for President Trump's expected summit with leader Kim in Singapore.
Kim will be the first senior North Korean official to visit the United States in nearly 20 years.
Mr Moon said he told Mr Kim that Mr Trump has a "firm resolve" to end hostile relations with North Korea and initiate economic cooperation if Kim implements "complete denuclearisation". The president noted Kim Yong Chol's visit, calling it a "solid response" to his letter. North Korean diplomats and officials are not allowed to travel more than a few miles outside New York City. Mr Kim Yong Chol, the vice-chairman of North Korea, heading now to NY.
"No matter how experienced and knowledgeable these officials are, they will not be able to change the fundamental challenge between the United States and North Korea over its status as a nuclear power".
"What's more important than anything from the latest inter-Korean summit was that the leaders easily got in contact, easily made an appointment and easily met to discuss urgent matters, without complicated procedures and formalities, just like a casual meeting", Moon told a meeting with senior secretaries.
The second round of talks lasted for about four-and-a-half hours, according to the report. The US has banned its citizens from entering North Korea, which has a history of detaining US citizens.
The Trump administration has demanded that North Korea completely and irreversibly shut its nuclear weapons program.
And on Tuesday, it was revealed that a top North Korean official widely known as leader Kim Jong Un's right-hand man had arrived in Beijing, en route to the United States for a meeting with top USA officials.
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The North Korea Leadership Watch website described Kim Yong Chol as sarcastic, hard to work with, and somewhat independent because it said he is "not properly deferential to his superiors". He said there is a "lot of good will" and denuclearization of the Korean peninsula would be "a great thing".
With much of the information about ongoing discussions at Panmunjeom withheld, what the countries are still working out appears to be the first steps they would take on the path to denuclearization - how much North Korea proactively dismantles its nuclear weapons programs, and how much rewards the U.S. can grant to the North in return, they say.
The North Korean envoy was travelling to the U.S. after speaking to Chinese officials in Beijing, South Korea's Yonhap news agency said.
Kim Yong Chol - a vice chairman of the Central Committee of the Workers' Party of Korea - is the highest-ranking official to visit the United States in nearly two decades.
Prime Minister Abe has expressed firm support for the US throughout the ups and downs of the summit-planning process, confirming last Friday that he supported Trump's surprise decision to withdraw the June 12 summit.
"It remains to be seen whether Kim Yong-chol's trip to the U.S. is to shake up the game as things are not working out, or to finalize details as the countries have reached an agreement".
Beijing backs the North's proposal for a phased and synchronized denuclearization process, while the United States insists on a comprehensive one-shot deal in which North Korea eliminates its nukes first and receives rewards later.