Moon, a liberal who took office in 2017, has lobbied hard to set up diplomacy between Trump and Kim to help find a peaceful settlement of the North Korean nuclear crisis.
Huge security has been deployed for the summit, with parts of the city-state in virtual lockdown.
President Donald Trump said Sunday a U.S. team was in North Korea to plan a summit with North Korean leader Kim Jong Un, raising expectations that the on-off-on meeting would indeed take place.
Trump's cancellation of the summit blindsided treaty ally South Korea, which had brokered the remarkable detente between Washington and Pyongyang.
Friday's response by North Korean Vice Foreign Minister was more conciliatory, specifically praising Trump's efforts for a meet, something which none of his predecessors have ever tried before.
North Korea to destroy nuclear site ahead of US summit; Donald Trump thanks Kim Jong Un for 'gracious' pledge
Dialogue brokered by Seoul has seen US-North Korea relations go from trading personal insults and threats of war last year to a summit between Kim and Trump due in Singapore.
Singapore's foreign ministry on Thursday confirmed it would host a historic meeting between US President Donald Trump and North Korean leader Kim Jong Un next month.
US President Donald Trump and North Korea's Kim Jong Un are likely to meet in Singapore next month, reports said Monday, as anticipation builds for unprecedented talks between the mercurial leaders.
Kim is set to meet South Korea's president Moon Jae-in on Friday, an event seen as a major stepping stone towards the summit with Trump.
Trump has had a series of telephonic conversations with a number of world leaders, who have been involved with the US on the North Korean issue.
The tweet also said the "time and place" of the historic meeting between Kim and Trump have yet to be decided.
Will there be a breakthrough? Or is the Trump-Kim summit merely the start of another long and difficult process meant to remove the North's nuclear capabilities?