Moon has been quoted as calling the recent agreements "akin to a declaration to end the Korean War", and has also expressed the optimistic view that there would be no more war forthcoming between North and South Korea.
A North Korean soldier defected to South Korea on Saturday across their heavily fortified land border, which the two sides have begun to demilitarise as relations between the Cold War-era foes warm, the South's military said.
Speaking to reporters Saturday en route to New Zealand, the final stop on his three-nation tour, Moon said Trump asked him to relay the message to Kim if and when he visits Seoul by the year's end as previously agreed by the leaders of the two Koreas.
After deriding Kim previous year as "little rocket man" and threatening his country with "fire and fury", Trump has bestowed Kim with praise.
In September Trump drew applause from a crowd of supporters at a campaign rally by describing "beautiful" letters he had exchanged with Kim, saying: "We fell in love, ok?"
Moon said he and Trump agreed during their meeting that Kim's trip to Seoul would play a "very positive role" in U.S.
Moon's spokesperson Kim Eui-Kyeom appreciated Trump's comment on second meet with Kim and said, "We hope a concrete agenda and logistics will be determined soon".
The Koreas and the U.S. -led U.N. Command recently finished removing firearms and troops from the jointly controlled area at the border village of Panmunjom, and eventually plan to allow tourists to freely move around there.
Moon said Kim's visit exclusively "depends on his decision".
After sharply raising tensions with nuclear and missile tests a year ago, Kim abruptly reached out to Moon and Trump with a vague nuclear disarmament pledge.
For his part, Trump discussed his intention to have a second U.S. -DPRK summit, and restated with Moon their commitment to closely coordinate on next steps. -North Korea summit in June.
Both the leaders has signed a roughly worded document over the denuclearization of the Korean peninsula, but the progress has been ceased as Pyongyang and Washington argued over the document's meaning.
"I think it will be absolutely imperative in this next summit that we come away with a plan for identifying all of the weapons in question, identifying all the development sites, allowing for inspections of the sites and the plan for dismantling nuclear weapons", Pence said in November.