The exact date and location of their next meeting have yet to be decided, but at their historic April summit in Panmunjom they agreed Moon would visit Kim in Pyongyang during the autumn.
Today's high-level talks, taking place on the northern side of the truce village in the Demilitarized Zone, were proposed by the North last week as it lashed out at Washington for pushing ahead with sanctions.
"We will review overall progress on carrying out the Panmunjom Declaration and discuss next steps," Unification Minister Cho Myoung-gyon, who is leading the South's delegation, told reporters ahead of the meeting.
"Views will be exchanged as well with regard to the autumn summit meeting agreed upon in the declaration," he added.
Cho addressed the possibility of Pyongyang raising the issue of sanctions to the South, and said: "We will explain our position to the North." The rapid rapprochement between the two neighbours that began this year paved the way for a landmark meeting between Kim and US President Donald Trump in Singapore in June.
Cross-border exchanges between the two Koreas have significantly increased since then, with the neighbours planning to hold reunions for war-separated families next week for the first time in three years.
But even as ties have improved, little progress has been made on the key issue of the North's denuclearisation.
Although Trump touted his summit with Kim as a historic breakthrough, the nuclear-armed North has since criticised Washington for its "gangster-like" demands of complete, verifiable and irreversible disarmament.
Meanwhile the US has urged the international community to maintain tough sanctions on the isolated regime.
If the third Moon-Kim summit takes place, the two are also expected to focus on hammering out a consensus on officially ending the 1950-53 Korean War, which concluded with an armistice instead of a peace treaty.
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