Seoul's Blue House says it's "carefully" watching the developments between Washington and Pyongyang.
Trump had said earlier Friday that the potentially historic summit might be getting back on track.
Trump had earlier indicated the summit could be salvaged after welcoming a conciliatory statement from North Korea saying it remained open to talks.
Moon and Kim held their first summit meeting on the South Korean side of Panmunjom on April 27.
"We'll see what happens".
"If it takes a war to end that nuclear threat, so be it".
In an apparent peace concession just before Trump canceled the summit, North Korea claimed to have destroyed key parts of Punggye-ri, its sole nuclear test site. Kim has said a formal end to the hostilities, along with a pledge from the USA not to attack his nation, would essentially eliminate Pyongyang's need for a nuclear arsenal.
Trump announced on Thursday that he was cancelling the planned meeting with Kim, which had been slated for June 12 in Singapore.
"Everybody plays games. You know that", he told reporters when asked about the ongoing talks.
A White House team will leave as scheduled for Singapore this weekend to prepare for the possible summit, a White House spokeswoman said.
Moon said Trump's decision left him "perplexed" and was "very regrettable".
Up until about two weeks ago, things seemed to be moving ahead as planned.
"Though North Korea's provocative statements led to [Trump's] decision, I don't think this is the time for us to be pessimistic or discouraged", Choo said, according to Yonhap.
Another U.S. delegation was in Singapore to work on logistics.
Trump accused North Korean leader Kim Jong Un of showing "open hostility" toward the Washington, adding that US nuclear weapons are far superior to North Korea's - and "so massive and powerful that I pray to God they will never have to be used". Trump took to his Twitter on Friday (local time) to confirm the occurrence of the summit.
The leaders of the states have signed the Panmunjom Declaration for peace, prosperity and unity in the Korean Peninsula.
SOuth Korea's President Moon Jae-in hugs with North Korea's leader Kim Jong-un.
Unlike last month's summit, which was held in front of live TV cameras, Saturday's meeting took place in utmost secrecy, with reporters only being told after the face to face had taken place.
"The first meeting would not solve all, but solving even one at a time in a phased way would make the relations get better rather than making them get worse", North Korean Vice Foreign Minister Kim Kye Gwan said in a statement carried by the North's official Korean Central News Agency, which mainly targets an external audience. Trump could envision Nobel laurels.
But in recent weeks the summit had been thrown into doubt by increasingly bellicose rhetoric from both top United States administration officials and Pyongyang.