The invitation was verbally delivered by Kim's sister, Kim Yo Jong, during a Saturday meeting at Moon's presidential compound in Seoul a day after the opening ceremony for the Winter Olympics in South Korea.
Kim Yo Jong, the powerful sister of North Korean leader Kim Jong Un, was among a high-level delegation in the stands as the joint Korean women's ice hockey team took on Switzerland in their opening match.
Even Japanese Prime Minister Shinzo Abe, who has shared the American skepticism of warming North-South relations and pressed Moon against falling for the North's "smile diplomacy", greeted Kim Yong Nam.
As well as the North Korean leader's sibling, Moon met the elderly Kim Yong Nam, the North's ceremonial head of state, during the appointment at the Blue House.More news: Several feared trapped after Taiwan quake
But away from the spotlight, these two unwanted guests, men dressed as North Korea supreme leader and the United States president, were escorted away from the premises.
Kim Yo-jong carried a bright blue folder as she entered the conference room at the Blue House, Seoul's presidential office, and some analysts speculated it could contain a letter from her brother.
The official also noted that while Trump believes in talking, in this case, the issue is not about words but about deeds on North Korea's part. Individual athletes were invited to participate as "neutrals" if they could prove they had not used performance-enhancing drugs.
Organising committee spokeswoman Nancy Park said the defence ministry and a cyber security team were investigating the outage that occurred Friday night.More news: Tech's Josh Pastner faces sexual assult lawsuit
The Koreans are rank outsiders in the tournament for which they only qualified by virtue of being the host nation and are likely to suffer defeat against the Swiss.
While a summit in Pyongyang would signal warming ties on the peninsula, it also risks driving a wedge in the alliance between the USA and South Korea.
"I leave here very confident that we are going to continue to do the things we've known have to be done to continue to pressure North Korea to abandon their nuclear ambitions", Mr. Pence said on Air Force Two.More news: Midwest pounded by heavy snow
Otto Warmbier died in June 2017, 18 months after he was taken in to North Korean custody for allegedly stealing a poster.
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