"The leaders of North Korea and the U.S. need to map out for a bigger picture and make a bold decision to take denuclearization further and to make the North dismantle the nuclear weapons," he said at a cabinet meeting on Tuesday, according to the presidential office.
The statement comes when concerns are growing over the deadlock in denuclearization talks between the U.S. and North Korea after the canceled trip by U.S. Secretary of State Mike Pompeo to Pyongyang.
"North Korea has to destroy their nuclear weapons and the U.S. has to respond with corresponding measures. The two countries need to remove deeply rooted distrust, built for the past 70 years of hostile relations," he said.
The White House said it is planning a second summit after Trump received a "very warm, very positive" letter from Kim and it is "already in the process of coordinating" the summit, according to White House press secretary Sarah Sanders.
Meanwhile, Moon and Kim agreed to meet for a third summit next week in Pyongyang.
Moon said the upcoming summit will focus on reducing military tensions and years of hostile relations between South and North Korea and the U.S. and North Korea.
He added that he is tasked with requests by the North and the U.S. to mediate differences between them and vowed to play the role.
"Complete denuclearization on the Korean Peninsula has to be achieved by negotiations between the North and the U.S.," he said.
The U.S. has demanded that the North should report a list of nuclear weapons first as the first step for denuclearization while the North wants to declare the official end of Korean War as a step for "lasting peace."
Moon sent special envoys to Pyongyang last week to set the date for the third inter-Korean summit and discuss ways to resolve the current stalemate in denuclearization.
At a cabinet meeting, Moon also called for a bipartisan support, in response to mixed reactions from South Korean lawmakers over Moon's invitation to the third summit in Pyongyang.
The presidential office announced on Monday it's inviting key members of the National Assembly, including the speaker and leaders of major parties.
Leaders of conservative opposition parties have rejected the invitation while liberal parties, including Moon's Democratic Party of Korea, agreed to accompany Moon to Pyongyang.