The most-recent meeting came after Trump canceled a planned June 12 summit with Kim, but then later said the United States was having "productive talks" about reinstating the summit.
North Korean state-run media KCNA said in its own statement about the Saturday meeting that Kim and Moon expressed "their stand to make joint efforts for the denuclearization of the Korean peninsula".
However, he hinted Friday that the meeting could happen after all and would "likely" happen on the originally scheduled date.
The two leaders held a surprise meeting at the Demilitarized Zone (DMZ), CNN reported, citing, the South Korean presidency, as saying in a statement.
The press secretary however, declined to elaborate further and said details of the summit would be revealed Sunday by President Moon himself. "That tells you how much Koreans want peace on the peninsula".
"Unlike what the Failing and Corrupt New York Times would like people to believe, there is ZERO disagreement within the Trump Administration as to how to deal with North Korea.and if there was, it wouldn't matter".
In response to Trump's cancellation, North Korea struck a more conciliatory tone, saying it was "willing to give the USA time and opportunities" to reconsider talks.
In particular, Tokyo is anxious to learn what North Korean leader Kim Jong Un said about denuclearization on the Korean Peninsula.
- The two Koreas will hold military talks to ease tensions, and Red Cross talks for reunions of families separated by the Korean War.
But Kim, through two summits with Chinese President Xi Jinping in March and May, has called for a phased and synchronized process in which every action he takes is met with a reciprocal reward from the United States.
Japanese Prime Minister Shinzo Abe said he respected and supported the USA president's move to cancel the summit while China, Pyongyang's sole major ally, urged the two foes to "show goodwill".
Last month's summit marked the first meeting between North and South Korean leaders in a decade.
Trump's decision to pull out of the summit with Kim came just days after he hosted Moon in a White House meeting where he openly cast doubts on the Singapore meeting but offered no support for continued inter-Korean progress, essentially ignoring the North's recent attempts to coerce the South.
The statement was issued in response to Trump's abrupt cancellation of the June summit between the two countries in the USA bid to get North Korea to abandon its nuclear weapon.
The summit, set for June 12, was jeopardized when, on May 11, the US and South Korea started a two-week annual air combat regional exercise, viewed by Pyongyang as a threat to its security.
South Korea's government said in a statement released Saturday that it was relieved about the revived talks for a summit.
But within 24 hours he reversed course saying it could still go ahead after productive talks were held with North Korean officials.
Moon said on Sunday: "Our journey to the complete denuclearization of the Korean Peninsula and a permanent peace agreement can not be stopped, and we have made a decision to cooperate closely".