"The meeting confirmed the determination of the parties to intensify the political and diplomatic process to resolve the entire complex of problems of the Korean Peninsula, including the nuclear one".
Now, after Kim has become the first North Korean leader to set foot on South Korean soil, Trump's planned meeting is increasingly important, with some pundits suggesting that the pair could sort out an action-ready plan for North Korean denuclearization and general peace on the peninsula before the year's end.
Buoying prospects for the Trump-Kim summit was a North Korean announcement on Sunday that it would close its main nuclear test site next month.
Kim could offer more disarmament concessions during his meeting with Trump, expected in May or June, but it's unclear what specific steps he would take.
Mr Moon's Nobel Prize comment came in response to a congratulatory message from Lee Hee-ho, the widow of late South Korean President Kim Dae-jung, in which she said Mr Moon deserved to win the prize in recognition of his efforts, the Blue House official said.
Moon has previously credited Trump with making a "huge" contribution to bringing the North and South together for the landmark talks.
Skeptics, however, say that Kim might be duping Moon and Trump into a false peace.
"I can only say 'so far, ' but he's talking about getting rid of the [nuclear testing] site".
It has yet to be seen how Kim will follow up on his promise to seek a peaceful solution to its nuclear and missile programs. "And he keeps it for a long time - longer than anyone has ever seen", added trump. "I may go in, it may not work out, I leave", Trump told the rally-goers.
Trump said Monday that "there's something that I like" about meeting at the border known as the Demilitarized Zone "because you're there".
While Trump said "numerous countries" are in the mix, last week he said only two or or three sites are being considered.
He warned against trust in U.S. President Donald Trump because "Iran's 40-year experience shows that the U.S. can not be trusted" and "does not live up to its global obligations".
Many other countries want the summit because "it has the chance to be a big event", Trump said.