Following his stint with the American delegation at speedskating, Moon attended a hockey game with Kim Yong Nam - the nominal North Korean head of state - on Saturday night.
Also at that game: the famed cheer squad from North Korea - more than 100 identically dressed women who sing, perform choreographed moves and cheer on their team.
But the verbal message on Saturday to come at a "convenient time" from dictator Kim Jong Un, delivered by his visiting younger sister, Kim Yo Jong, is part of a sudden rush of improving feelings between the rivals during the Pyeongchang Olympics.
The US has been cautious about the invitation, with vice president Mike Pence also in South Korea to warn the country against "falling for" the North's overtures.
Her presence helped the theme of Korean unity to dominate the opening ceremony as the two Koreas marched under one flag.
The meeting could be a prelude to a summit between leaders from the two countries, and Moon has already said he is willing to travel to Pyongyang to meet Kim Jong-un.
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.
Douglas Paal, a former senior US diplomat under previous Republican administrations, said North Korea held the propaganda edge for now because, "it's tough not to get caught up in the emotions of an Olympics event".
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Japan does not participate in the military exercises, but it is within reach of North Korean missiles and relies heavily on US forces and their readiness to deal with that threat. The North Korean orchestra will return home on Monday after their second and last performance on Sunday.
They were on both sides of the ice, chanting back and forth. It was the first time an inter-Korean team had competed at an Olympic Games. The U.S will not allow the propaganda charade by the North Korean regime to go unchallenged on the world stage.
A senior Japanese defence official said North Korea's Games charm offensive could "simply be a way to gain time" until it completed its nuclear and ballistic missile development. Saturday's game ended in a crushing 8-0 loss to Switzerland.
Sunday's 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".
By also sending a youthful, photogenic individual who would surely draw worldwide attention at the Olympics, Kim might have also been trying to construct a fresher image of the country, particularly in face of USA efforts to use the Olympics as an occasion to highlight the North's brutal human rights record.
Moon gave qualified consent to holding a future summit with Kim, the first between the two Koreas since 2007. The previous meetings came after rounds of worldwide talks aimed at eliminating the North's nuclear program, which eventually failed. Reviving inter-Korean dialogue is critical for the policies of Moon, who insists that Seoul should be in control in global efforts to deal with the North Korean nuclear issue.
In South Korea and beyond, commentators are taken with Kim's surprisingly stylish and relaxed demeanor.
But analysts say it may be more hard for the South to arrange a summit with the North coming off a year in which Pyongyang test-fired dozens of missiles, including three ICBMs, and conducted its most powerful nuclear test to date.