Significant differences, however, remain - especially over the US and North Korean definition of "complete denuclearization".
Should the reports of working-level talks be officially confirmed, Kim will be the most high-ranking North Korean official to visit the US and hold talks with government officials since the late Vice Marshal Jo Myong Rok's meeting with then-President Bill Clinton in 2000.
Moon could travel to Singapore for a three-way summit with Kim and Trump next month, Yonhap reported Monday. -North Korea summit are continuing to move forward. Moon said on Sunday that Kim requested the meeting, only the fourth ever by leaders of the two countries since the Korean War.
Andrew O'Neil, an expert on North Korea and a professor at Griffith University in Australia, called the visit a "significant development".
General Kim Yong-chol, 72, has emerged as a key player in North Korean diplomacy in the first half of 2018.
The two diplomats have crossed paths several times in the past, especially during the six-party negotiation process.
The United States and South Korea blacklisted Kim Yong Chol for supporting the North's nuclear and missile programmes in 2010 and 2016, respectively.
Kim Yong-chol, North Korea's former spy chief, would become the highest ranked official from the isolated nation to visit the United States since 2000, when Pyongyang sent Vice Marshal Jo Myong-rok to meet then-President Bill Clinton.More news: Oil ministry gives in-principle nod to launch petrol, diesel futures: ICEX
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The talks are spearheaded by Sung Kim, current U.S. ambassador to the Philippines and former special envoy to the six-party talks on denuclearization of North Korea.
The White House said Trump and Japanese prime minister Shinzo Abe talked on the phone on Monday and confirmed they would meet before the "expected" US-North Korea summit.
As well as Kim Yong-chol's visit, Kim Jong-un has also sent his de facto chief of staff, Kim Chang Son, to Singapore via Beijing.
North Korea claims to have destroyed its primary nuclear test site.
In Kim and Moon's first, upbeat meeting on April 27, they agreed to seek the "complete denuclearisation" of the Korean peninsula - but didn't define what that meant, or how that would proceed. The US National Security Council's Korea specialist Allison Hooker and Randall Schriver, assistant secretary of defense for Asian and Pacific security affairs, are also part of Sung Kim's team.
The State Department said a separate team of White House officials has headed to Singapore to sort out logistics. The president did not give further detail of the talks, but the general is expected to meet Secretary of State Mike Pompeo, whom he had already encountered during his visit to Pyongyang earlier this month.
American and North Korean officials then met at the North-South Korean border on Sunday.
There will be no similar media events at the village until after June 12, the possible date of the summit between Trump and Kim Jong Un, according to the command.
During her stay in Pyongyang, Kim Jong Il took Albright to one of the mass games spectacles that included a giant mosaic displaying a rocket flying into the sky.