Each Koreas agreed final month to denuclearise the area, at a border summit which got here after months of warlike rhetoric from Kim and Trump.
KIM Jong-un has warned the United States about utilizing "strain and army threats" in opposition to it forward of historic peace talks.
And last week, Mr Trump said Washington's strength is going to keep us out of nuclear war. Earlier, he said details of the summit are being ironed out, and that he'd spoken with the leaders of and Japan about preparations for the meeting.
Besides Singapore, analysts have speculated that the venue for the Trump-Kim meeting could also be Mongolia, which Kim could travel to by bulletproof train.
Moon's office said Sunday that he and Trump agreed in their phone call that the US president should meet with Kim as soon as possible in a bid to maintain "the momentum of the success" of Friday's historic meeting of the Korean leaders.
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French President Emmanuel Macron urged Trump to stay in the deal when he met with the president in Washington last month.
Meanwhile, Seoul's presidential security advisor Chung Eui-yong was joined by his American counterpart John Bolton Friday in dismissing a New York Times report claiming that the U.S.is considering a troop reduction in South Korea.
They will also discuss the USA president's own upcoming summit with the North Korean leader, a statement said.
At the meeting, the two leaders are expected to discuss the issue of denuclearization of North Korea.
He also criticized Washington for its ongoing "pressure and military threats" and its position that such pressure won't be eased until North Korea gives up its nuclear weapons completely.
Some hardline USA advisers to President Trump, including the newly appointed senior security adviser John Bolton, would try their best to cast doubts in the accords struck between the two Korean leaders, and even help derail them altogether as non-dependable. He made a bold gesture with the invitation to North Korea to send athletes and a delegation to the Winter Olympics, held in South Korea earlier this year.