President Moon Jae-in sent Chung Eui-yong, head of the National Security Office, to China and Suh Hoon, director of the National Intelligence Service, to Japan on Monday after they returned from the United States, where they met with Trump and delivered Kim's invitation to a North-U.S. summit.
But that hasn't stopped officials, analysts, and other observers from debating the pros and cons of possible summit sites, ranging from North Korea's capital city of Pyongyang, to the Joint Security Area (JSA) between the two Koreas, to farther afield in other areas of Asia or Europe. A nuclear North Korea could spark a chain reaction, with Japan, South Korea and even Taiwan seeking to obtain their own nuclear weapons.
"I'm very anxious about that happening again and I would hope that this administration understands that", he said.
The Japanese leader said Tuesday that he welcomes any dialogue premised on denuclearization of North Korea.
Some experts are concerned that North Korea could be seeking to delay worldwide action while strengthening its nuclear program.
The U.S. and South Korea hold military maneuvers every year. Kim is likely to raise the possibility of such a treaty along with establishing diplomatic relations and moving toward nuclear disarmament, during a meeting with the USA leader, the Dong-A Ilbo newspaper said Monday, citing an unidentified senior official in South Korea's presidential office. That doesn't seem to be the case here - this South Korean-brokered meeting has come somewhat out of the blue.
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On Saturday, Trump said his talks with Kim could end with no agreement or they could be "the greatest deal for the world".
Sweden has held unofficial talks in the past, and North Korea's foreign minister is expected to visit soon, according to Swedish media.
Observers say the meeting has raised expectations of progress in resolving the nuclear crisis on the Korean peninsula.
He added: "Should we succeed, there will be dramatic changes in world history, and (South) Korea will have played the leading role". But they warn the process leading to removal of all nuclear weapons from the area is complex.
Chung announced last week that Trump said he would meet Kim by May "to achieve permanent denuclearization" of the Korean Peninsula.
The Security Council has imposed tough economic sanctions aimed at choking off revenue to Pyongyang's military programs after Kim's regime carried out a sixth nuclear test and a series of advanced missile launches.
Trump's aides have been wary of North Korea's diplomatic overtures because of its history of reneging on worldwide commitments and the failure of efforts on disarmament by previous United States administrations.