Trump had called off the meeting on Thursday before backtracking a day later, saying the summit could still take place.
He added that if necessary, the meeting "will be extended beyond that date".
The statement was issued in response to Trump's abrupt cancellation of the June summit between the two countries in the US bid to get North Korea to abandon its nuclear weapon.
Bolton had suggested in April that the USA could follow the "Libya model of 2003, 2004", in which Libyan leader Muammar Gaddafi abandoned his nuclear program in exchange for sanctions relief before being killed in 2011 during a US -backed uprising.
"I wish to place a great meaning on the latest talks that were held as if they were an ordinary event between friends", he said.
Report has it that the two leaders spoke at length for two hours about the possible and potential of the June 12 meet between Kim and Trump still happening. But there's also skepticism whether Kim will ever agree to fully relinquish his nuclear arsenal, which he likely sees as his only guarantee of survival.
Moon's office said a press conference is set for Sunday morning in Seoul.
Protesters hold candle lights during a rally to denounce the United States' policies against North Korea near the US embassy in Seoul, South Korea, Friday, May 25, 2018.
Trump now says he remains open to holding a summit and noted "the lines of communication had opened between the two countries".
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The North's state-run KCNA news agency said the two leaders agreed to "meet frequently in the future to make dialogue brisk and pool wisdom and efforts, expressing their stand to make joint efforts for the denuclearisation of the Korean peninsula".
He has since said he is open to reinstating the talks. He characterized the recent back-and-forth as the "usual give and take".
It was the second time in a month the leaders of the two Koreas met, this time just north of the demilitarized zone separating the countries.
It wasn't immediately clear how the rivals organized what appeared to be an emergency summit.
Moon was blindsided by Trump's decision to abruptly announce he was canceling the summit, less a day after returning from meeting in the Oval Office created to keep the momentum in the diplomatic effort.
Moon was accompanied by his spy chief, Suh Hoon, while Kim was joined by Kim Yong Chol, a former military intelligence chief who is now a vice chairman of the North Korean ruling party's central committee tasked with inter-Korean relations. But on Saturday, the two leaders spoke face to face.
"The two leaders candidly exchanged views about making the North Korea-U.S. summit a successful one and about implementing the Panmunjom Declaration", South Korea's presidential spokesman said in a statement.
The announcement came as Trump took to social media to lambast a New York Times article which claimed that it was now too late for the summit to take place as originally scheduled. He made a quick decision to accept the meeting in March, over the concerns of many top aides, and has remained committed, even amid rising concerns about the challenges he faces in scoring a positive agreement.