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.
"I think we're going to do one, fairly, into January, February", Mr Trump told reporters aboard Air Force One as he returned late last Saturday from the G-20 summit in Argentina.
US Secretary of State Mike Pompeo was due to meet with a top North Korean official in early November, but the meeting was abruptly put off, with North Korea insisting that Washington ease sanctions.
The former South Korean foreign minister said on Monday "this is the time" to achieve Pyongyang's denuclearization after almost a quarter century of unsuccessful efforts, adding that the North and the world should not lose this opportunity.
Trump said earlier in October that his meeting with Kim would be held after the midterm Congressional elections, and the venue of his second meeting with Kim had been narrowed down to "three or four" locations.
Moon's programme in Auckland includes a ceremony of welcome and a state lunch hosted by Governor-General Dame Patsy Reddy at Government House, a wreath laying ceremony and tour of Auckland War Memorial Museum, and an engagement with the Korean community in Auckland.
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The South Korean daily Hankyoreh reported that Moon was angry about press reports claiming discord between the USA and South Korea.
Since the first summit in June, progress on North Korea's denuclearization appears to have stalled, with neither side making headway on key stumbling blocks. Also, working on his North Korea diplomacy, the president met with the leaders of the Netherlands and South Afirca.
"This will be the first time that a North Korean leader will visit South Korea so, although there is no time frame set for that, still that's very meaningful", Moon said.
North and South Korea have begun to remove landmines and destroy military bunkers at parts of their common border as part of efforts to improve long-strained relations. But prospects of a fourth Moon-Kim meeting have recently dimmed, with negotiations on denuclearizing the North grinding to a halt.
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.
Moon spoke shortly after a meeting with New Zealand Prime Minister Jacinda Ardern where they discussed how to secure a denuclearised Korean peninsula.