Following a whirlwind 24 hours that saw Trump cancel the highly anticipated June 12 meeting with Kim before saying it's potentially back on, the Korean leaders took matters into their own hands.
"Should the North Korea-US summit succeed, I would like to see efforts to formally end the (Korean) war through a three-way summit of the South, the North and the US", he said.
She also said it was up to the Americans whether they would "meet us at a meeting room or encounter us at nuclear-to-nuclear showdown".
"Chairman Kim and I have agreed that the June 12 summit should be held successfully, and that our quest for the Korean peninsula's denuclearisation and a perpetual peace regime should not be halted", Mr. Moon said.
"It appears that (the North) remains honest in implementing the agreement and making efforts on denuclearization and peace building", added the minister in charge of inter-Korean affairs.
They also reportedly talked "frankly" about how they could make the potential summit between Kim and Trump a success.
A White House team will leave as scheduled for Singapore this weekend to prepare for the possible summit, a White House spokeswoman said on Saturday.
The meeting was the latest dramatic turn in a week of diplomatic ups and downs surrounding the prospects for an unprecedented summit between the United States and North Korea, and the strongest sign yet the two Korean leaders are trying to keep the on-again off-again meeting on track.
Trump's original decision to abandon the historic summit initially blindsided South Korea, which had been brokering a remarkable detente between Washington and Pyongyang and is desperate to avoid conflict breaking out.
Moon and Kim held talks in the demilitarised zone separating their two nations on Saturday in a scramble to save the June summit.
North Korea's state news agency, KCNA, said the Korean leaders had agreed to "high-level" talks between the two countries on June 1.
In response, the North said Friday that it stays open to talks with the USA anytime and in any format.
A TV screen shows South Korean President Moon Jae-in, left, meeting with North Korean leader Kim Jong Un at the border village of Panmunjom during a news program at the Seoul Railway Station in Seoul, South Korea, May 26, 2018.
"Kim Jong-un told Moon Jae-in to positively cooperate with each other as ever to improve the DPRK-U.S. relations and establish mechanism for permanent and durable peace".
The main dispute between the US and North Korea boils down to how fast Kim should give up his weapons, and what he'll get in return.