The summit could in turn provide renewed momentum to talks over denuclearization between North Korea and the United States, after Trump cancelled a visit to the North by his secretary of state last month citing a lack of progress.
But despite the difficulties, Kim's "trust in Trump remains unchanged", Chung said.
A convoy of buses carrying participants of an inter-Korean family reunion makes its way past a South Korean guard post through the Demilitarized Zone (DMZ) towards North Korea, in Goseong on August 20, 2018. Kim dismissed such worries, Chung said, and told the South Korean delegation that a formal end of the Korean War would not be linked to the withdrawal of the U.S. troops.
Mr. Moon is seen as eager to keep the diplomacy alive in part so that he can advance his ambitious engagement plans with the North, which would need US backing to succeed.
Reversing the defiant statements of his officials during that period, Kim said he was willing to accept "stronger measures" to dismantle his nuclear program and pressed for a formal conclusion to the Korean War in return, a peace treaty regarded by strategic analysts as a significant boon to North Korea at this point.
President Moon, who had mediated the first US-North Korea summit held in Singapore in June, is expected to play a bigger role trying to push for a breakthrough in the impasse, so as to advance inter-Korea ties with blessings from its security ally US. But negotiations have made little headway.
Chung said Kim showed "frustration over the doubt raised by some parts of the worldwide community about his willingness to denuclearise, and asked us to convey his message to the United States".
He even expressed regret that the global community does not appreciate such meaningful steps taken by his country, the South Korean envoy said. The two leaders will discuss measures on denuclearization, establishing peace and detailed steps to ease military tensions, Chung said.
In addition to setting a date for their meeting in Pyongyang, Moon and Kim Jong Un also agreed to open a joint liaison office at the Gaeseong industrial complex north of the border before the summit, Chung said.
It will be the third meeting between the two leaders this year, as Moon makes a major effort to improve relations with the North in the hope this will convince Kim scale back or abandon his nuclear arsenal.
Soldiers march across Kim Il Sung Square during a military parade in Pyongyang.
Washington wants Kim to disclose his nuclear and missiles programmes and assets, and cancelled a planned trip to Pyongyang by Mike Pompeo, the USA secretary of state, late last month.
In this year's talks, North Korea and the United States have been at odds over whether denuclearization or declaring an end to the 1950-53 Korean War should come first.
"North Korea needs the declaration of the end of the war to convince its own officials of denuclearization", Koh said.
A peace declaration could put the U.S.in a tricky position.
News 1 reported Thursday the biggest US warship based in Asia could arrive in South Korea for joint exercises.
The United States, however, has said it is committed to building a peace mechanism if North Korea denuclearizes.
Pyongyang apparently wants to first achieve an official end to the seven-decade state of war with South Korea, while Washington wants to start immediately on the long process of denuclearization.