On Thursday Trump cancelled his upcoming meeting with Kim which had been due to take place in Singapore on June 12 - only to reverse course a day later and say it could still go ahead after productive talks were held with the North Korean oficials.
On Sunday, current United States ambassador to the Philippines Sung Kim led a team to the northern side of the "Unification House" along the DMZ for talks on North Korea's nuclear program.
Koh said Moon would try to persuade Kim to accept an alternative approach advocated by Seoul, in which the North's comprehensive commitment and credible actions toward denuclearization are followed by a phased but compressed process of declaration, inspection and verifiable dismantling.
U.S. officials said the personal attack on Pence and threat of nuclear war was the final straw.
"If it takes a war to end that nuclear threat, so be it".
South Korea, which brokered the talks between Washington and Pyongyang, was caught off guard by Mr. Trump's abrupt cancellation of the summit in which he cited hostility in recent North Korean comments. "Chairman Kim is anxious about whether he can trust that the USA will end the policy of hostility and guarantee the stability of his regime" after denuclearization.
Trump called off the meeting on Thursday, citing the "tremendous anger and open hostility", in recent statements from Pyongyang, and North Korea hardened its rhetoric toward the USA, warning it was ready for a "nuclear-to-nuclear" showdown if the US didn't follow through on the summit.
The Moon-Kim meeting, which came Just days after Trump canceled his planned June 12 summit with Kim in Singapore, was unexpectedly announced after it had already wrapped up. He told reporters Friday that "everybody plays games".
Analysts say Mr. Kim's diplomatic outreach in recent months after a flurry of nuclear and missile tests in 2017 indicates he is eager for sanctions relief to build his economy and the worldwide legitimacy the summit with Mr. Trump would provide. "We would like to do it", Trump said, and "they very much like to do it".
Sung Kim met with North Korea's Vice-Foreign minister Choe Son Hui, who said last week that Pyongyang was "reconsidering" the talks, the daily said.
Kim and South Korean President Moon Jae-in met in April in what was only the third meeting between the leaders of the two states. At the Pentagon, Defense Secretary Jim Mattis said "diplomats are still at work on the summit, possibility of a summit, so that is very good news". "We will soon see where it will lead, hopefully to long and enduring prosperity and peace. Only time [and talent] will tell!" the president tweeted. Later, Kim Jong-Un assembled the party's military commission or the first time in two years.
But as the clock ticks toward June 12, Trump's aides are describing a renewed sense of urgency to make progress that would allow the two leaders to meet on schedule. Kim Jong-un's younger sister Kim Yo-jong, who attended the first inter-Korean summit on April 27, escorted Moon before the summit. Trump could envision Nobel laurels.