In a briefing on Thursday (Sept 6) morning, South Korea's national security chief Chung Eui-yong, who headed a five-member delegation to the North on Wednesday, said the two leaders will examine the implementation of the Panmunjom Declaration agreement they made when they first met in April, and how to push the pact forward.
"(The countries) agreed to discuss specific ways to built their mutual trust and prevent armed conflicts while continuing their ongoing talks on reducing military tension at the South-North Korean summit", Chung said.
"Kim said that he would like to build on this trust to end the 70 years of hostilities between the North and the United States within Trump's first term, improve relations and achieve denuclearization", Chung said. And it won't until Pyongyang gives up its nuclear weapons, the US says. North Korea has taken several steps such as dismantling its nuclear testing site this year, but the US wants it to take more serious disarmament measures.More news: Trump is Ignorant, White House Dysfunctional
The two Koreas also agreed to hold a third summit September 18-20 in Pyongyang.
At a press conference, the envoys said Mr Kim told them it was his firm will to work with South Korea to achieve a lasting peace settlement. Kim wants to sign a treaty first, but American officials fear that he's after a propaganda win and will use the peace treaty to press for the removal of almost 30,000 US troops in South Korea.
North and South Korea remain technically at war as the 1950-53 Korean War ended in a ceasefire.
"Chairman Kim emphasized that he has never made a negative comment about Trump", Chung Eui-yong, South Korea's top national security adviser, said at a media briefing on Thursday.More news: Bob Woodward's book describes a dysfunctional Trump White House
That message will very likely contain an appeal for Washington to formally declare an end to the war - something Pyongyang says is vital to building trust and ending decades of hostility.
Kim and Trump held a landmark summit in Singapore in June, in which they agreed to work toward complete denuclearization.
Since entering talks this year, North Korea has taken several steps such as dismantling parts of its main rocket launch site and closing its nuclear testing site.
The United States, however, has said it is committed to building a peace mechanism if North Korea denuclearizes.
The war ended with an armistice, not a peace treaty, meaning US -led United Nations forces, including South Korea, are technically still at war with the North.More news: Tropical Storm Gordon brings hurricane warning for parts of Gulf Coast
After Trump and Kim signed a joint declaration in Singapore in June to work towards the denuclearization, in exchange for Washington guaranteeing Pyongyang's security, the White House has grown impatient over the absence of concrete measures from the regime.