Kim used his annual New Year address to warn he has a "nuclear button" on his table, but sweetened his remarks by expressing an interest in dialogue and participating in the Pyeongchang Games.
South Korea's unification minister Cho Myoung-Gyon told a press conference that Seoul was "reiterating our willingness to hold talks with the North at any time and place in any form".
"The government proposes to hold high-level government talks with North Korea on January 9 at the Peace House in Panmunjom," Cho said, referring to a truce village in the demilitarised zone separating the two Koreas.
"We hope that the South and North can sit face to face and discuss the participation of the North Korean delegation at the Pyeongchang Games as well as other issues of mutual interest for the improvement of inter-Korean ties."
Cho added that the specifics of the proposed talks, including its agenda, could be discussed through the inter-Korean hotline at Panmunjom, which has been cut off since 2016.
The Koreas, divided by the demilitarised zone since the end of the 1950-53 Korean War, last held high-level talks in 2015 in an effort to ease border tensions.
Those talks were led by then-South Korean national security adviser Kim Kwan-Jin and his North Korean counterpart Hwang Pyong-So but failed to reach an agreement.
"Just the fact that they are meeting will be meaningful because it signals an attempt on both sides to improve relations," said Koh Yu-Hwan, a political science professor at Dongguk University.
But once they sit down, the North could put Seoul in a difficult position by making unacceptable demands such as an end to the annual joint military drills with the United States, Koh added.
"What North Korea is trying to do is re-establish its relations (with Seoul) as a nuclear state. The South's dilemma is whether we can accept that," Koh said.
Dovish South Korean President Moon Jae-In, who has long favoured engagement to defuse tensions with the North, earlier Tuesday welcomed Kim's suggestion that there could be an opportunity to kick-start dialogue.
However, he indicated that improvements in inter-Korean ties must go hand in hand with steps towards denuclearisation of the North.
Moon proposed Red Cross and military talks last year, but Pyongyang did not respond.
North Korea has rattled the international community in recent months with multiple missile launches and its sixth and most powerful nuclear test -- purportedly of a hydrogen bomb.
Kim's comments on Monday were the first indication of North Korea's willingness to participate in the Winter Games, which run from February 9 to 25.
Moon called them a "positive response" to Seoul's hope that the Pyeongchang Olympics would be a "groundbreaking opportunity for peace" and urged officials to come up with measures to realise the North's participation.
The main venues for the Games are just 80 kilometres (50 miles) from the heavily fortified border with North Korea and the build-up to the event has been overshadowed by the nuclear weapons standoff.
In his speech Monday, Kim said the Olympics could provide a reason for officials from the neighbours "to meet in the near future".
"Since we are compatriots of the same blood as south Koreans, it is natural for us to share their pleasure over the auspicious event and help them," Kim said in his address.
Seoul and the Games' organisers are very keen for the North to take part.
Analysts say North Korea's participation at Pyeongchang is likely, given its leader's public remarks about sending a delegation there.
Two North Korean athletes -- pairs figure skaters Ryom Tae-Ok and Kim Ju-Sik -- qualified for the Games but Pyongyang's Olympic Committee missed an October 30 deadline to confirm to the International Skating Union that they would participate.
The pair could still be invited to compete by the International Olympic Committee.
North Korea's past participation in sporting events in the South has largely depended on the political and military situation, though they did send a full team to the 2014 Asian Games in Incheon, near Seoul.
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