The unannounced meeting at the Panmunjom border village between Moon and Kim came a month after they held the first inter-Korean summit in more than a decade at the same venue and declared they would toward a nuclear-free Korean peninsula.
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.
-North Korea summit a success, the presidential Blue House said Saturday.
Earlier the leaders of two Koreas had a historic meeting in Panmunjom, South Korea.
Less than a year after the two leaders traded threats of nuclear war, the back-and-forth over whether the summit will even happen reflects both Trump's lead-from-the-gut style of decision-making and North Korea's long-standing penchant for unpredictable behavior.
The talks are expected to continue tomorrow at Tongilgak, or "Unification House", the building in the northern part of the demilitarised zone where Kim met Moon on Saturday, The Post reported.
Coming after a surprise meeting between Kim and South Korean President Moon Jae-in, the move was the latest sign that the on-again, off-again Trump-Kim summit may yet go ahead as planned.
While peace between the warring North and South Koreas seemed feasible just weeks ago, hesitation by North Korean dictator Kim Jong Un to dismantle their nuclear weapons arsenal has prompted the US president to rescind his offer to host a peace summit in Singapore.
Photos released by the Blue House show Kim and Moon hugging and Kim's propaganda chief and sister, Kim Yo Jong, greeting Moon.
The North Korean government said Friday it is still "willing to give the USA time and opportunities" to reconsider talks "at any time".
Asked about the summit, Trump said Friday: "We'll see what happens, it could even be the 12th", referring to the June 12 date originally set. KCNA said the leaders also agreed to meet frequently.
Kim Jong Un agrees with me on this. The meeting between the two would be centered around, if all goes according to plan, the denuclearization of North Korea. "We're talking to them right now".
Trump blamed the cancellation on the "tremendous anger and open hostility" in a recent statement by the North Koreans.
A three-way summit was proposed by Moon and Kim when they first met at the border village of Panmunjom on April 27, Yonhap said.