Trump, speaking after welcoming an American who had just returned to the US after being freed from jail in Venezuela, said that working level meetings with the North are ongoing at a location "not so far away from here" and they have "gone very very well".
Moon and Kim held a historic summit in April where they announced vague aspirations for a nuclear-free peninsula and permanent peace, which Seoul has tried to sell as a meaningful breakthrough to set up the summit with Trump.
USA and North Korean officials have disagreed about the method for denuclearizing the Korean Peninsula, with the US calling for complete, immediate, and irreversible denuclearization, while Pyongyang advocated for a gradual dismantling of the nuclear program in exchange for security and economic guarantees along the way.
South Korean President Moon Jae-in said earlier that he and North Korea's Kim agreed at a surprise meeting on Saturday that the possible North Korea-U.S. summit must be held, Moon told a news conference in Seoul.
South Korean president Moon Jae-in told reporters that he was "very perplexed by the situation", and that "it is very regrettable that the North Korea-U.S. summit will not be held on June 12 when it was scheduled to be held", according to Yonhap News Agency.
Video and one of the photos released by the presidential Blue House on Saturday showed Kim hugging Moon and kissing him on the cheek three times as he saw Moon off after their meeting at Tongilgak, the North's building in the truce village, which lies in the Demilitarized Zone (DMZ), the 160-mile (260 km) long, 2.5-mile (4 km) wide buffer that runs along the heavily armed military border.
Trump pulled out of the summit on Thursday, citing Pyongyang's "tremendous anger and open hostility" - only to reverse himself on Friday after North Korea said it was willing to talk "at any time".
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South Korea believes North Korea remains committed to improving relations despite strongly criticizing Seoul over ongoing U.S. Not surprisingly, North Korea bristles at the mention of Libya . "Nothing has changed on North Korea that we know of", Mr.
"The two leaders exchanged their candid views on the implementation of the April 27 Panmunjom Declaration and successful opening of the North-US summit", the statement said.
In a tweet, Mr Trump said the meeting could still take place on 12 June in Singapore "and if necessary will be extended beyond that date". "North Korea dialogue. We are continuing to watch developments carefully".
Mr Trump's aides had warned that merely agreeing to the summit had provided Mr Kim with long-sought worldwide legitimacy and, if Mr Trump ultimately backed out, he risked fostering the perception that the president was insufficiently committed to diplomatic solutions to the nuclear question. "We'll see what happens- it could even be the 12th". "It's moving along very nicely".
Just hours before Mr Trump cancelled the summit, North Korea said it had carried out its promise to dismantle tunnels at its only nuclear test site.
Moon expressed "deep regret" over the summit's cancellation and urged that direct talks between Washington and Pyongyang be held as soon as possible.
By Friday, North Korea had issued a statement saying it was still "willing to give the USA time and opportunities" to reconsider talks "at any time, at any format". It was the first meeting between leaders of the two countries in 10 years.