Moon and Kim held a surprise meeting on Saturday at the border village of Panmunjom, during which they agreed that a North Korea-US summit must be held, Reuters reported.
"What's more important than anything from the latest inter-Korean summit was that the leaders easily got in contact, easily made an appointment and easily met to discuss urgent matters, without complicated procedures and formalities, just like a casual meeting," Moon told a meeting with senior secretaries.
Trump last week pulled out of the meeting with Kim, planned for June 12 in Singapore, before announcing he had reconsidered and American and North Korean officials were meeting to work out details.
On Sunday, the US State Department said US and North Korean officials had met at Panmunjom, a village in the Demilitarized Zone (DMZ) that runs along the heavily armed border between North and South Korea.
A "pre-advance" team of US officials was also traveling to Singapore to meet with North Koreans there, the White House said.
A US official said that Sung Kim, the former US ambassador to South Korea, was leading the American delegation to meet North Korean officials at the border.
"It’s a good thing to have him onboard," said a former senior South Korean official who worked with Sung Kim in past. "He’s capable, level-headed, cautious, and has solid grasp of the issues and knows North Koreans well. But at the same time he has healthy skepticism."
North Korea has faced years of economic sanctions over its nuclear and missile programs since it conducted its first nuclear test in 2006.
The United States has struggled to slow the country's weapons programs, given Pyongyang's promise to develop a nuclear-tipped missile capable of hitting the US mainland.
In remarks on Sunday, Moon acknowledged Pyongyang and Washington may have differing expectations of what denuclearization means and he urged both sides to hold working-level talks to resolve their differences.
The United States has demanded the "complete, verifiable, and irreversible" dismantlement of North Korea's nuclear weapons program. Pyongyang has rejected unilateral disarmament and has always couched its language in terms of denuclearization of the Korean Peninsula.
In previous, failed talks, North Korea said it could consider giving up its arsenal if Washington removed its troops from South Korea and withdrew its so-called nuclear umbrella of deterrence from South Korea and Japan.
Moon said Saturday's summit with Kim, which was organized on short notice after the North Korean leader requested a meeting, should be a model for increased contact between the leaders of the two Koreas.
"If we could hold working-level, back-to-back talks on both sides of Panmunjom if urgently necessary in addition to formal summits, it would expedite faster advancement of inter-Korean relations," Moon told his aides.