Moon's spokesman Kim Eui-kyeom said Saturday Kim's sister verbally delivered his offer in a lunch meeting with Moon at Seoul's presidential palace.
Moon met Kim Yo Jong - a close confidante of her brother and the first member of the dynasty to set foot in the South since the Korean War - and the North's ceremonial head of state Kim Yong Nam at the Blue House in Seoul. Between the lunch and talks, they spent nearly three hours together. Sunday's demonstration was held near a theater where Kim Jong-un's sister Kim Yo Jong and President Moon Jae-in were going to watch a performance by Pyongyang's Samjiyon Orchestra, part of the hundreds-strong delegation the North sent down for the Olympics. "It's not unfamiliar", Kim said when asked by one of the South Koreans at the dinner how she feels on the other side of the border.
Moon's office says Kim's sister, Kim Yo Jong, came to the South as his special envoy.
As North Korea has made peaceful overtures towards the South ahead of the games, it's thought to be unlikely it's behind whatever attack hit the Games' website.
This is in line with Moon's hopes that the inter-Korean detente could lead to broader negotiations on the core challenges facing the Korean Peninsula.
Kim Yo Jong and Kim Yong Nam arrived Friday afternoon, leading a senior delegation.
Even the official Olympics Twitter page is throwing it back to 20 years ago at the opening ceremony in Japan.
BH Loya death case: Opposition delegation submits memorandum to President
Also, the memorandum read that "Such a procedure will ensure the credibility of the institution in the eyes of the public". The matter was leading to yellow journalism, Rohatgi said.
The United States insisted it would continue its policy of exerting global pressure on North Korea in an effort to get it to abandon its nuclear program. Instead of eating with Moon and the other dignitaries, Pence ate with the American athletes. Those photos quickly went viral.
The two nations have marched together at an Olympics but this time have taken the extraordinary step of combining on the composition of the women's hockey team.
Pence was scheduled to leave for the United States on Saturday night.
Cybersecurity researchers said in January they had found early indications that Russia-based hackers may be planning attacks against anti-doping and Olympic organisations in retaliation for Russia's exclusion from the Pyeongchang Games.
Any meeting would represent a diplomatic coup for Moon, who swept to power previous year on a policy of engaging more with the reclusive North and has pushed for a diplomatic solution to the standoff over North Korea's nuclear and missile program.
However, few officials have said this in quite the colorful language that Trump has used.
And second, USA and North Korean officials this high up the food chain rarely if ever get this close to one another - let alone in the midst of a tense nuclear standoff of the kind we've seen over the past several months.
Mr. Moon gave qualified consent to holding a future summit with Mr. Kim, the first between the two Koreas since 2007.