Kim Yo Jong made the comments today during a luncheon hosted by Moon at Seoul's presidential palace where she delivered her brother's invitation for a summit meeting in Pyongyang.
Seoul's presidential office says Moon during the luncheon proposed a toast calling for peace and "mutual prosperity" for the two Koreas.
Moon recalled his past visit to the North's Diamond Mountain resort, where he and his mother met his North Korean aunt during a temporary reunion of families separated by the 1950-53 Korean War.
Moon's office says Moon, sister Kim Yo Jong and North Korea's nominal head of state Kim Yong Nam are at the Kwandong Hockey Center today to watch the game.
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Kim Yo Jong and Kim Yong Nam are on an unprecedented visit to South Korea amid abrupt reconciliation mood between the rival Koreas for the Pyeongchang Winter Olympics.
The North Koreans earlier today invited Moon to visit Pyongyang in what would be the Koreas' third summit talks since their 1945 division.
Aides did not immediately say whether the invitation was discussed as the two sat next to each other viewing several race heats today.
They were seated not far from a section of North Korean cheerleaders who are attending the Olympics in a sign of warming ties between the two Koreas that Pence and other US officials have warned against.
Moon and Kim Yong Nam, the North's ceremonial head of state, will jointly attend the unified Korean women's ice hockey team's first match later this evening.