The talks between the two Korean leaders 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.
"We're doing very well in terms of the summit with North Korea", Trump was quoted as saying by Reuters at the White House.
Mr Trump said in a letter to Kim on Thursday he was cancelling the planned Singapore summit, citing North Korea's "open hostility".
South Korean President Moon Jae-in shakes hands with North Korean leader Kim Jong Un during their summit at the truce village of Panmunjom, North Korea, in this handout picture provided by the Presidential Blue House on May 26, 2018.
U.S. officials crossed into North Korea on Sunday to hold talks on preparations for a possible summit, a U.S. newspaper reported, as North Korean leader Kim Jong Un reaffirmed his commitment to meet with U.S. President Donald Trump.
May 22: South Korean President Moon Jae-in and Trump met in Washington.
Moon said that the USA has an understanding of the North's sincerity toward denuclearization talks, highlighting that the two countries agreement to hold a summit shows that Washington has confirmation of Kim's intentions.
Trump past year launched what his administration has called a "maximum pressure" campaign against North Korea combining the toughest-ever US and global economic sanctions with diplomatic actions and the Republican president's military threats and preparations.
Moon was accompanied by his spy chief, Suh Hoon, while Kim was joined by Kim Yong Chol, a former military intelligence chief who is now a vice chairman of the North Korean ruling party's Central Committee tasked with inter-Korean relations.
However, separately on Saturday, White House Press Secretary Sarah Sanders revealed that a team of U.S. officials was being dispatched to Singapore for summit preparations.
USA officials had said, after Trump's announcement that the summit was off, that North Korean officials had not been responding to their communications.
"While Washington and Pyongyang have expressed their hopes for a summit through published statements, Moon has to step up as the mediator because the surest way to set the meeting in stone would be an official confirmation of intent between heads of states", Hong said. When asked about the talks between Moon and Kim, the president replied, "the talks have gone very well".
May 15 (evening EDT): North Korea's state media made a statement under Vice Foreign Minister Kim Kye Kwan's name.
The two leaders embraced as Moon departed.
In addition to his summits with Moon and Xi, Kim also has had two meetings with U.S. Secretary of State Mike Pompeo.
"Even though they share the same resolve, there need to be discussions regarding the roadmap for how to make it happen, and that process could be tough", he said.
Ahead of that summit, the Koreas established their first-ever leaders' hotline to enable Moon and Kim to engage in direct communication and defuse crises. Moon said Trump's decision left him "perplexed" and was "very regrettable".
South Korea was caught off guard by President Trump's abrupt cancellation of his summit with Kim, with the USA president citing hostility in recent North Korean comments. Military talks and talks on meetings of divided families are also scheduled.
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