"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.
Trump has frequently described a warm personal relationship with Kim, arguing that this rapport would help him succeed at a diplomatic breakthrough that has eluded US presidents since the 1950s.
A North Korean soldier defected to South Korea on Saturday across their heavily fortified land border, which the two sides have begun to demilitarise as relations between the Cold War-era foes warm, the South's military said.
However, North Korea has taken steps to cease its ballistic missile and nuclear programs.
On Friday, South Korea's Defense Ministry said both militaries had completed the removal of 20 border guard posts and a number of landmines from an area in which they plan to conduct joint searches for the remains of soldiers killed in the 1950-53 Korean War.
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The statement said Xi and Trump "agreed that great progress has been made with respect to North Korea".
A USA think tank said last month it had identified at least 13 of an estimated 20 active, undeclared missile bases inside North Korea, underscoring the challenge for American negotiators hoping to persuade Kim to give up his weapons programs.
They agreed on the need for "maintaining vigorous enforcement of existing sanctions to ensure the DPRK understands that denuclearization is the only path", Sanders said, using the North´s official name, the Democratic People´s Republic of Korea. "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".
Differences have emerged between Washington and Seoul on how to proceed with Kim, as the dovish Moon has long favoured engagement with the North.
Trump, speaking aboard Air Force One on his return from Argentina, tells reporters that he and Kim are "getting along very well" and have a "good relationship". The South Koreans have expressed hope the visit - which would be a first for a North Korean leader - could happen by the end of this year.
The South Korean government is optimistic over Kim's visit to Seoul, which he promised during the third inter-Korean summit in September in Pyongyang.