"The message is that President Trump has very favorable views toward Chairman Kim and he likes him", Moon told reporters aboard a flight from Argentina to New Zealand, where he started a three-day state visit on Sunday. He said, "We are getting along very well".
Mr Trump has frequently described a warm personal relationship with Mr Kim, arguing that this rapport would help him succeed at a diplomatic breakthrough that has eluded USA presidents since the 1950s.
"President Trump discussed his intention to have a second U.S. -DPRK summit", spokeswoman Sarah Sanders said, using the acronym for the North's formal name the Democratic People's Republic of Korea.
The postponement stirred concerns that nuclear talks between the USA and North Korea had again bogged down, as Kim's regime seeks sanctions relief before taking key disarmament steps.
To improve long-strained relation between the South and North Korea, they both have started their mission by removing landmines and also destroying military bunkers at their border.
The two leaders signed a vaguely worded document on denuclearisation of the peninsula, but progress since has stalled as Washington and Pyongyang spar and North Korea has taken few concrete steps to abandon its nuclear and ballistic missile programs.
On Friday, a South Korean train left for North Korea, launching an 18-day inspection of railways there as part of efforts to eventually connect rail lines over the border, a project their leaders agreed on earlier.
Mr Moon's comments came one day after he met the US President at the G20 economic summit in Argentina. Ban said he hopes the summit will occur at a time when the worldwide community feels confident about North Korea's commitment to denuclearization.
They have also begun work to reconnect a train line and fix another rail link across the border.
Former UN Secretary-General Ban Ki-moon has urged North Korean leader Kim Jong Un to take concrete steps toward complete denuclearization and gain the trust of the global community if he wants sanctions lifted.
Despite the warming ties, it remains unclear whether Kim will make his first-ever visit to the South this year, as Seoul is hoping.
In September, the President said that he wanted to schedule a second summit with Kim.