Moon said, "We need to offer water-tight guarantees" for Kim's safety during any visit.
Moon spoke to reporters on Saturday aboard his presidential plane en route to New Zealand from Argentina, where he met Trump on the sidelines of a Group of 20 Nations summit.
Kim sharply raised tensions with nuclear and missile tests last year, but suddenly reached out to South Korea and the United States this year with a vague nuclear disarmament pledge. He said, "We are getting along very well".
Meanwhile, Trump said a second summit with Kim could take place in January or February.
A train carrying South Korean engineers and officials crossed into the North on Friday to begin a landmark joint survey to reconnect railway tracks between the two Koreas. -North Korea nuclear diplomacy.
President Moon Jae-in's office quoted Moon as saying that Trump asked him to convey those messages to the North Korean leader if he visits Seoul this year as he promised.
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North Korea had entered into agreements with regional powers in 1994 and in 2005 to dismantle its nuclear program in return for economic benefits and diplomatic rewards, but those deals broke down after Pyongyang clandestinely continued to pursue building weapons of mass destruction.
The White House said in a statement on Saturday following Trump's meeting with Chinese President Xi Jinping that the US and North Korea would strive "to see a nuclear-free Korean Peninsula". But critics say the South risks conceding some of its conventional military strength before North Korea takes any meaningful steps on denuclearization, as the larger nuclear negotiations between Washington and Pyongyang seemingly drift into a stalemate.
The South's dovish President Moon has long favoured engagement with the nuclear-armed North and has dangled large investment and joint cross-border projects as incentives for steps towards denuclearisation.
Prospects for a second summit to overcome a current logjam in negotiations will likely depend on whether either the US or DPRK can show flexibility with regards to offering gestures surrounding sanctions relief or steps towards denuclearization.