Moon and Kim agreed at a surprise second meeting on Saturday that a possible North Korea-U.S. summit, now planned for June 12 in Singapore, must be held successfully, Moon told a news conference in Seoul.
Meanwhile, Trump spoke to reporters about the potential summit from the Oval Office on Saturday night, where he had gathered with lawmakers to celebrate the return of Utah man Joshua Holt, who had been jailed in Venezuela for almost two years and finally freed.
Mr Trump's decision to pull the summit is understood to have followed comments from North Korean diplomat Choe Son Hui, who referred to Mike Pence as a "political dummy". President Trump said on Friday that the summit might even be held on the originally scheduled date of June 12.
"There are meetings going on as we speak", Trump said. He said there is a "lot of good will" and denuclearisation of the Korean Peninsula would be "a great thing".
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Mr Trump had rattled the region on Thursday by cancelling his meeting with Mr Kim, citing "open hostility" from Pyongyang.
But North Korea's propaganda took a very aggressive line with the United States since Wednesday, threatening nuclear war and bashing Vice President Mike Pence.
A senior South Korean official later said that the two Koreas are discussing a possible non-aggression pledge and the start of peace treaty talks as a way of addressing Pyongyang's security concerns ahead of U.S.
"(Kim) once again has made clear his will for the complete denuclearization of the Korean Peninsula and expressed his intent to settle the history of war and confrontation, and to cooperate for peace and prosperity through the success of the 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. -North Korea summit on track, said he delivered Trump's "firm will" to end the hostile relationship with North Korea and pursue bilateral economic cooperation.
"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.
American officials are skeptical that Kim will ever fully abandon his nuclear arsenal, and Moon said North Korea is not yet convinced it can trust security guarantees from the United States. At that meeting, they declared they would work toward a nuclear-free Korean peninsula and a formal end to the 1950-53 Korean War. It allowed Moon to push for a U.S. "That hasn't changed", the USA leader said. "I urge North Korea and the U.S.to confirm each other's will by sharing their problems with each other and communicating directly".
But within 24 hours he reversed course, saying it could still go ahead after productive talks were held with North Korean officials.