North Korean leader Kim Jong Un's invitation to Mr Moon for a summit in Pyongyang has also left Tokyo anxious.
Kim's sister Kim Yo Jong, who is visiting South Korea as his special envoy for the Winter Olympics, conveyed her brother's invitation during a meeting at the presidential Blue House, officials from the Blue House said.
South Korea's presidential office said after the meeting that the North Korean leader has invited Moon to visit his country and expressed willingness to improve inter-Korean ties in the letter.
Camera flashes; mobs of reporters: Through it all, Kim has maintained a placid smile - and despite Pence's stated intention to reinforce North Korea's isolation during his Olympic visit, the sister of the repressive regime's leader seems to have spent at least as much time with Moon as Pence did.
The North showed off long-range missiles, including the Hwasong-15 intercontinental ballistic missile that may be capable of reaching the USA mainland, in a message that it would not abandon its nuclear and missile programs.
But the North Korean cheerleading squad there to support the team?
Most notably, Kim sat directly behind U.S. Vice President Mike Pence at Friday night's opening ceremony in Pyeongchang, creating the unique image of two top officials whose governments have been locked in a loud argument, suddenly seated just feet away from each other. Police are shown trying to stop demonstrators from burning portraits of North Korean leader Kim Jong-un during an anti-North Korea protest before the opening ceremony for the Pyeongchang Winter Olympics on Friday. I assume they knew that message was coming.
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His speech (viewable below) attracted major media attention in the United States, with ESPN's hugely popular Sportscenter program labelling it an "epic rant".
Tensions in the Korean peninsula have cooled down in recent weeks thanks to the North Korean regime's decision to send its athletes to take part in the Olympic games. Kim Jong-nam, the leader's half brother, was killed in Malaysia a year ago, apparently on the orders of the Pyongyang regime.
Moon told Kim Yo Jong that the North and South should continue to build conditions for a summit, Moon spokesman Kim Eui-kyeom said.
The KCNA report cited Moon as saying inter-Korean relations should be mended by the parties concerned "at any cost as indicated by Chairman Kim Jong Un in his New Year Address".
Kim Yong Nam, the North's nominal head of state who was also at Saturday's meeting, said "even unexpected difficulties and ordeals could be surely overcome and the future of reunification brought earlier when having a firm will and taking courage and determination to usher in a new heyday of inter-Korean relations".
Angela Ruggiero, a four-times ice hockey world champion and Olympic gold medallist, told Reuters she would ask others to nominate the team, which included 12 players from North Korea which is still technically at war with the South.
However, South Korea's main opposition party warned that any talks between the two Koreas where the scrapping of North Korea's nuclear program was not a precondition would only "benefit the enemy".
Pence is playing "right into North Korea's hands by making it look like the USA is straying from its ally and actively undermining efforts for inter-Korean relations", said Mintaro Oba, a former diplomat at the State Department specialising in the Koreas, who now works as a speechwriter in Washington.
"What can not be overlooked is that Moon did not directly demand North Korea to abandon its nuclear development".