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.
'As such, he asked me to tell Chairman Kim that he wants to implement the rest of their agreement together and he will fulfil Chairman Kim's wishes'. Trump told reporters travelling home to Washington with him aboard Air Force One from Argentina that "three sites" were in consideration for the meeting, a follow-up to their historic summit in Singapore in June.
Moon said that he and Trump agreed that the sanctions should not be lifted at this point (although the recent agreement to allow a train-crossing of the demilitarized zone to test the tracks in the North constituted an "exception" to the sanctions, which could be a precedent for future similar "exceptions").
Moon emphasised the impact Kim's presence in South Korea would have on the two nations.
"We remain unchanged in our stance that (Kim's visit) will be possible and necessary before the end of this year", Baik Tae-hyun, a foreign ministry spokesman, said in a press briefing.
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Moon has also facilitated a series of high-level talks between the United States and North Korea, including a summit between Trump and Kim in June.
Asked if the meeting would take place in the U.S., Trump said that it may happen at "some point", although diplomats are now looking at three other locations for the next summit. The two countries signed a free-trade deal in 2015. Also, working on his North Korea diplomacy, the president met with the leaders of the Netherlands and South Afirca. But both Moon and Trump say fears of war on the Korean Peninsula have eased sharply.
North and South Korea have begun to remove landmines and destroy military bunkers along parts of their common border amid efforts to improve long-strained ties.
Trump recently said he is likely to hold a second summit with Kim in January or February.
Three sites are being considered, he added.
They agreed on the need for "maintaining vigorous enforcement of existing sanctions to ensure the DPRK understands that denuclearization is the only path", Sanders said, using the North's official name, the Democratic People's Republic of Korea.