"President Trump has a very friendly view of Chairman Kim and he wishes Chairman Kim would implement the rest of their agreement and that he would make what Chairman Kim wants come true,"...
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 message is that President Trump has very favourable views towards Chairman Kim and he likes him", Mr Moon told reporters aboard a flight from Argentina to New Zealand, where he started a three-day state visit on Sunday. "We have a good relationship", Mr Trump told reporters aboard Air Force One on his return from the G-20 summit.
Moon said there still is a possibility Kim's visit could take place before the end of the year.
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.
On Friday, Trump discussed the situation with South Korean President Moon Jae-in on the sidelines of the G20 summit.
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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.
Kim agreed to travel to Seoul after hosting Moon in Pyongyang in September for their third summit this year. But both Moon and Trump say fears of war on the Korean Peninsula have eased sharply. North Korea has taken few concrete steps to abandon its nuclear and ballistic missile programs.
North and South Korea have begun to remove landmines and destroy some military bunkers along their common border as part of the detente.
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.
Ban said he hopes their second summit will occur at a time when the worldwide community feels confident about North Korea's commitment to denuclearisation.
"They have not lived up to the commitments so far", Bolton said at a Wall Street Journal's CEO Council conference in Washington, according to CNN.