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.
United States officials insist on the complete, verified and irreversible denuclearisation of the Korean Peninsula before sanctions are lifted.
"We have actually talked about three sites, we haven't determined the sites", he said. Moon and his wife, Kim Jung-sook, arrived in the South Pacific nation late Sunday after attending the Group of 20 Nations summit in Argentina. The initial summit took place in June in Singapore but Trump has said the next meeting will probably take place at a new location. But both Moon and Trump say fears of war on the Korean Peninsula have eased sharply.More news: Kylie Jenner's Latest Instagram With Travis Scott Is Raising Eyebrows
Ban, a former South Korean foreign minister, said "this is the time" to achieve North Korea's denuclearisation after almost a quarter century of unsuccessful efforts, and that the North and the world should not lose this opportunity.
The two leaders signed a vaguely worded document on denuclearisation of the peninsula, but progress since has stalled as Washington and Pyongyang spar over the meaning of the document.
Meanwhile, Bloomberg has reported Mr Trump as saying he expects to hold a second summit with Mr Kim as soon as next month.More news: Wolves without Ruben Neves and Jonny Otto for Chelsea visit
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.
Trump signalled on Saturday he was likely to meet Kim again in January or February.
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. Moon said it's still unclear whether Kim will visit Seoul by the end of this year, and that it's up to the North Korean leader. But prospects of a fourth Moon-Kim meeting have recently dimmed, with negotiations on denuclearizing the North grinding to a halt. The two leaders said Kim's visit to the South Korean capital "would provide additional momentum to their joint efforts to establish peace on the Korean peninsula", Moon's press secretary Yoon Young-chan said.More news: Jon Rahm wins in Bahamas, ends year with 3 titles
Mr Moon said that he would carry a warm message from Mr Trump to Mr Kim, who may visit Seoul as early as this month.