Both of the summits, one in 2000 and the other in 2007, took place when South Korea was under liberal presidents who sought rapprochement with North Korea despite conservative criticism that they provided the North with little-strings-attached aid shipments.
The North's call came after the two Koreas agreed Tuesday, at high-level talks on the reclusive country's participation in the upcoming Winter Olympics in the South, to hold military talks to reduce tensions.
The prospects of resolving decades-old tensions on the divided Korean Peninsula remain deeply uncertain.
Speaking on the bilateral talks, which have so far resulted in the North agreeing to send a delegation of athletes to the Olympics along with a cheering and performance team, Moon said, "The once completely blocked channel of communication with the North has now been reopened". Along the way, six-country talks aimed at dismantling North Korea's nuclear program became moribund.
The United States has also warned that all options, including military, are on the table in dealing with the North.
Plans to bring a North Korean team to the Winter Olympics here have run into a problem: Providing financial or other forms of support to the North could put the South in violation of worldwide sanctions.
North Korea encouraged this meeting with a warm, New Year's message to the South. South and North Korea will now have to bring their views in line as much as possible before the four-party meeting with the International Olympic Committee on January 20. He played down fears of war.
Pence is expected to be joined by first daughter Ivanka Trump, but her attendance isn't confirmed, officials told the Wall Street Journal. "A lot of good talks are going on right now". Although I see value in North Korea participating in the Olympics, our past dealings with them tell me that it should not be on their terms only. "I'm a very flexible person".
However, in the lead-up to the 2014 Asian Games, South Korea said it would follow worldwide norms that stipulate attending countries should pay their own way.
Trump suggested that his fiery tweets against North Korea and Kim during the first year of his administration were part of a diplomatic strategy.
The North's reaction to these proposals is not known.
Working talks will be held soon to work out the details of bringing the North Koreans to the Olympics, the statement said, with the exact schedule to be decided via documented exchanges.
The current uncertainty comes at a time when senior officials admit time is running out until Trump is faced with a dilemma over whether to try to destroy the North Korean threat to the USA mainland, or to live with it.
It was a radically different tone from the rhetoric of recent months, which have seen Kim and US President Donald Trump trade personal insults and bellicose threats of war, while Pyongyang has launched missiles capable of reaching the US mainland and carried out its sixth and most powerful nuclear test to date.
"The two leaders underscored the importance of continuing the maximum pressure campaign against North Korea". "It's going to include everything from the size of the visiting group to detailed matters concerning the joint South and North Korea entrance procedures", Sung said. At the 2002 Asian Games in South Korea's southern port city of Busan, a large ship moored in the harbor served as North Korea's lodging.
An Chan-Il, a defector researcher who runs the World Institute for North Korea Studies, said the cheerleaders are cherry-picked by the regime based on tough criteria.
Information for this article was contributed by Josh Wingrove and other staff members of Bloomberg News.