Moon said Kim saw the June 12 summit in Singapore as a pivotal opportunity to end decades of confrontation.
But both sides have since been working to get it back on track.
A TV screen shows South Korean President Moon Jae-in, left, meeting with North Korean leader Kim Jong Un at the border village of Panmunjom during a news program at the Seoul Railway Station in Seoul, South Korea, May 26, 2018.
Trump signalled that preparations for the summit with Mr. Kim were going ahead.
"And if President Trump's brand, and that's the right word here, going into this meeting demands something like that, this is going to end up in a very bad place".
The visit by the United States delegation was first reported by the Washington Post later on Sunday.
"He crossed the line that separates the two Koreas to meet with Choe Son Hui, the North Korean vice foreign minister who said last week that Pyongyang was "reconsidering" the talks", WaPo reported.
"During the South Korea-US summit, President Trump said the US is willing to clearly put an end to hostile relations (between the US and North Korea) and help (the North) achieve economic prosperity if North Korea conducts denuclearisation".
It allowed Mr Moon to push for a US-North Korean summit that he sees as the best way to ease animosity that had some fearing a war a year ago.
While China would nearly certainly block any US attempt to win passage of more U.N. sanctions, the United States still has at its disposal further unilateral measures such as imposing its own new sanctions on North Korean officials and entities or additional Chinese companies that do business with Pyongyang.
What came out of Saturday's surprise talks?The two leaders embraced as Moon departed.
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Moon has insisted Kim can be persuaded to abandon his nuclear facilities, materials and bombs in a verifiable and irreversible way in exchange for credible security and economic guarantees.
Kim said the two Koreas should "positively cooperate with each other as ever to improve the DPRK-US relations and establish mechanism for permanent and durable peace", KCNA added, saying South and North Korea would hold another round of "high-level" talks on June 1.
"We're talking to them now", Trump said of the North Koreans.
"Kim Jong Un knows his [nuclear] program inside and out", Hayden told ABC News Global Correspondent and "This Week" co-anchor Martha Raddatz on Sunday.
The flurry of diplomatic backslapping and bonhomie disappeared in recent weeks as the summit was thrown into doubt by increasingly bellicose rhetoric from both top U.S. administration officials and Pyongyang. On Saturday, they met just 24 hours after a phone call from Pyongyang. Moon was accompanied by National Intelligence Service Director Suh Hoon, and Kim Jong-un by Kim Yong-chol, vice chairman of the central committee of the ruling Workers' Party. He said he expects the talks to go smoothly because Pyongyang and Washington both know what they want from each other.
US President Donald Trump points to members of the media as he arrives on the South Lawn of the White House in Washington, Friday, May 25, 2018, after attending a graduation and commissioning ceremony at the US Naval Academy in Annapolis, Maryland.
In addition to the U.S. exit from the Iranian nuclear deal, Macron's talks with Putin are set to focus on the conflicts in Syria and Ukraine. The US has banned its citizens from entering North Korea, which has a history of detaining US citizens.
Mr Trump has not ruled out such an incremental approach.
But a rapprochement began in January when Mr Kim suggested he was "open to dialogue" with South Korea.
Moon said the results of Saturday's summit had been communicated to the United States.
Moon Jae-in holds press conference on May 27.