When asked Saturday if he would ever host Kim in the United States, the Republican president replied: "At some point, yeah".
"President Trump and I share a view in that regard".
"The crisis is still continuing because of North Korea's development of nuclear, missile and weapons program and materials".
Former UN Secretary-General Ban Ki-moon has urged North Korean leader Kim Jong-un to take concrete steps toward complete denuclearization if he wants sanctions lifted.
Differences have emerged between Washington and Seoul on how to proceed with North Korea, as Moon has long favoured engagement with the North. "It is North Korea which should clearly state and show by taking actions, then I am sure that there will be no reason why the Security Council of the United Nations will continue to impose the sanction on North Korea", Ban said.
Mr Trump, who met Mr Kim in Singapore in June, said on Saturday that he is likely to meet the North Korean leader for a second time in January or February, with three sites for their meeting under consideration.
North and South Korea have begun to remove landmines and destroy some military bunkers along their common border as part of the detente.
"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.
But prospects of another meeting in coming weeks have dimmed as denuclearisation talks have stalled, prompting Moon to say that the planned visit's timing should not overshadow its historic nature.
Moon said he and Trump agreed during their meeting that Kim's trip to Seoul would play a "very positive role" in US-North Korea nuclear diplomacy.
Moon is now pushing for Kim to visit Seoul before the end of the year, in what would be the first official visit from a North Korean leader to the capital since the end of the Korean War.