US President Donald Trump and South Korean President Moon Jae-in say tensions have eased significantly since then, but Ban, who is in Tokyo for a World Bank event, said the crisis is not over and that North Korea should fully disclose its nuclear weapons inventory to a trusted global organisation such as the worldwide Atomic Energy Agency.
The South Korean daily Hankyoreh reported that Moon was angry about press reports claiming discord between the USA and South Korea. 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.
North Korea is seeking security guarantees from the U.S. and relief from worldwide sanctions.
South Korea's president said Tuesday no timeframe has been set for a historic visit to Seoul by North Korean leader Kim Jong Un despite both hoping it will take place this year.
"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.
Ban said he hopes their second summit will occur at a time when the global community feels confident about North Korea's commitment to denuclearisation.
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 made the comments to reporters on the final day of a visit to New Zealand.
After Trump and Kim met in Singapore in June, negotiations between Washington and Pyongyang have stalled due to differences on the scale of denuclearization, US sanctions, and whether to issue a war-ending declaration, among others.
He said such a visit would "accelerate the denuclearisation in the Korean peninsula" and result in a "greater stride in relations improvement" between the North and the United States. "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.
Moon said a visit by Kim to the South would also improve US-North Korea relations as Kim and President Donald Trump look to follow up their June summit in Singapore.
Trade is also likely to feature, following New Zealand and South Korea signing a free trade agreement in 2015.