Trump welcomed Kim's remarks in a trademark Tweet.
Mr Chung is also expected to brief countries including the USA on the results of his visit to Pyongyang, and "closely work together with them". "The United States seems likely to move if North Korea gives something more concrete", he said.
Still, the South Korean envoy quoted the North Korean leader as saying he hoped to completely denuclearize and end the hostile relationship between his country and the USA before Trump's first term ends in January 2021.
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 USA backing to succeed.
"There is still an enormous amount of work to do", he said.
They are (from L) Yun Kun-young, presidential secretary for state affairs; Vice Unification Minister Chun Hae-sung; Moon's special envoy and head of the presidential National Security Office Chung Eui-yong; Kim; National Intelligence Service Director Suh Hoon; Kim Sang-gyun, a senior NIS official; and Kim Yong-chol, a senior member of the North's People's Party.
At a news conference aired on South Korean TV and reported by the South Korean news agency Yonhap, the envoy, Chung Eui-yong, said the meeting - the third between the two leaders - would be September 18 to 20 in Pyongyang. Picture taken September 5, 2018.
North Korea's official KCNA news agency said Kim told the South's envoys that his "fixed stand" was to turn the Korean peninsula into "a cradle of peace without nuclear weapons, free from nuclear threat".
Kim emphasised that his "trust in Trump remains unchanged" despite the difficulties, Chung said, and expressed his intention to work closely with the United States to achieve denuclearisation "in the first official term of President Trump".
"He said he would appreciate that such good faith is accepted with good faith", Chung said.
Most observers warn the North has taken no meaningful steps to denuclearise.
In a move that will irk North Korea, the U.S. Justice Department in Washington was preparing Thursday to announce charges against a North Korean in connection with the hack of Sony Pictures Entertainment in 2014, according to a U.S. government official familiar the criminal case. Post-summit nuclear negotiations between Washington and Pyongyang were rocky and quickly settled into a stalemate.
Relations between the North and the U.S. have soured since the historic Trump-Kim summit in Singapore in June. But negotiations have made little headway, while signs North Korea has maintained work on its weapons have emerged.
"The United States shouldn't delay any further an end-of-war declaration, which the US president promised at the Singapore summit", the North's official Rodong Sinmun newspaper said in an editorial.
"The discussions at the summit will also be centered around issues on the permanent settlement of peace and common prosperity on the Korean Peninsula as well as, particularly, practical measures to realize denuclearization on the Peninsula", he said.
"This was meant to be set up in August but delayed because of the troubled negotiations between Pyongyang and Washington". Dongguk University professor Koh Yu-hwan said it was significant that Kim has finally committed to a timetable for denuclearization, even though he was vague on exactly what he planned to do by the end of President Trump's first term.