Trump surprised the South Koreans in March by hastily agreeing to meet Kim in a summit, the first such meeting between a sitting American president and a North Korean leader.
Moon said both he and Kim had "agreed the June 12 North Korea-U.S. summit must be successfully held".
Moon told reporters Sunday that Kim "again made clear his commitment to a complete denuclearisation of the Korean Peninsula", and that he told the South Korean leader he's willing to cooperate to end confrontation and work toward peace for the sake of the successful North Korea-US summit.
"The White House pre-advance team for Singapore will leave as scheduled in order to prepare should the summit take place", White House press secretary Sarah Sanders said. "It's moving along very nicely", he said.
Trump tweeted earlier Saturday that a summit with Kim, if it does happen, will likely take place on June 12 in Singapore as originally planned.
He told reporters Friday that "everybody plays games", and said: "They very much want to do it, we want to do it, we'll see what happens".
"At the talks there were in-depth exchanges of opinions to tackle the matters which should be resolved to quickly carry out the Panmunjom Declaration agreed upon at the third North-South summit and to realize the denuclearization of the Korean peninsula and achieve regional peace, stability and prosperity", KCNA said.
While maintaining that Mr. Kim is commited to denuclearisation, Mr. Moon acknowledged Pyongyang and Washington may have differing expectations of what that means, and he urged both sides to hold working-level talks to resolve their differences. When asked about the talks between Moon and Kim, the president replied, "the talks have gone very well". There has been no response yet to Saturday's developments in Korea from official United States channels.
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Cruz also told the pope of his "most wonderful friends, who are also Catholic and who struggle with this". You have to be happy with who you are", Cruz reportedly told El País , a Spanish newspaper.
South Korea's President Moon Jae-in speaks during a news conference at the presidential Blue House in Seoul on Sunday.
Trump rattled the region on Thursday by cancelling his meeting with Kim, citing "open hostility" from Pyongyang.
A senior USA official says the North Koreans stood up an American delegation last week seeking to make arrangements for the summit and did not return messages from the US seeking to discuss the meeting.
Pyongyang's harsh rhetoric toward Washington came after Kim Jong-un visited Dalian, China from May 7 to 8 and held his second summit with Xi.
On Sunday, a team of US officials visited North Korea to help prepare for the summit. "And there's no more concrete sign that this is going forward than having that group of, that team of three, as reported by the Washington Post, sitting down with the North Koreans to hammer out those details", said Lee. "And Kim wants legitimization, of course - meeting President Trump - and also, he needs a counterweight to the Chinese".
"Chairman Kim requested a meeting without any formality two days ago and I gladly accepted it", Mr. Moon said, noting that the summit was in line with previous agreements for the two leaders to meet more often.
The main dispute between the US and North Korea boils down to how fast Kim should give up his weapons, and what he'll get in return.
Saturday's talks were only the fourth time serving leaders of the two Koreas, who remain technically at war, have ever met.