During their meetings, Kim made light-hearted remarks about how he is viewed outside North Korea in global media and elsewhere, one Blue House official said. The envoy was led by a veteran diplomat with close ties to the US, National Security Advisor Chung Eui-yong.
"There is good reason to talk, but only if we are talking about something that is worth doing and that could be reasonably verified", said former Defense Secretary William Perry, who dealt with North Korea during President Bill Clinton's administration.
What do the North Koreans want?
People watch a television news report showing pictures of President Trump, left, and North Korea leader Kim Jong Un at a railway station in Seoul on March 9.
A senior administration official told reporters the president deserves credit for the dramatic developments. Chung was flanked by Suh Hoon, the head of South Korea's intelligence agency, who was also at the dinner in Pyongyang, and Cho Yoon-jae, the South Korean ambassador to the United States. The communist leader has previously said the US and South Korea must end their joint military exercises before he would agree to talks.
Joseph Yun, the USA envoy in charge of negotiating with North Korea, quit last week, and Trump has yet to nominate an ambassador to South Korea.
"The May meeting will be recorded as a historic milestone that realized peace on the Korean Peninsula", he said.
"Pyongyang can say they tried and resume nuclear and missile testing", he said. "He does the easy things, and if he can get him at Mar-A-Lago, even better".
What does the United States want?The European Union, Russia and China - whose leader spoke by phone with Trump on Friday - have all welcomed the move.
"Just an extraordinary evening, and of course opening the door to the big question: If President Trump can truly solve this problem, that would be going down as a great president, and there's no way around that", Erin Burnett said on her show OutFront. In fact, North Korea is a puppet of the Chinese. But Pyongyang had stuck to its hardline stance, not agreeing to come to the table for talks on denuclearization. Co-host Willie Geist said it's unclear whether the planned meeting would actually come to pass or if it was part of a larger negotiation over the rogue regime's nuclear weapons.
But analysts were damning, saying a US-North Korea summit was a win for Kim.
South Koreans responded positively to the news, with online comments congratulating Moon for laying the groundwork for the Trump-Kim talks.
Mr Davis said President Trump must push North Korea to take steps towards denuclearisation before any meeting takes place. "And I think they are right to argue that nothing is going to change just because there is going to be a meeting", Rubio said.
"We should as Christians be praying about this meeting".
At the dinner on the first night, South Korean officials were able to observe the relationship between Kim and his wife, Ri Sol Ju, said the second administrative source.
He stressed that China would continue to maintain sanctions on North Korea until a political settlement was reached.
"(Trump) absolutely needs to get there", Biggs said.
There are weeks of work ahead to sort out the logistics of a potential meeting.
During President Barack Obama's administration, North Korean officials were also seeking a breakthrough with the United States, and were disappointed when American officials offered no diplomatic concessions, former U.S. director of national intelligence, James Clapper, has said.
"A Trump meeting with Kim presents both risks and opportunities", said Bonnie Glaser, an Asia expert at the Center for Strategic and International Studies think tank. "At that point, he will go to the world and say 'I'm very reasonable".
Tillerson did not define the precise difference between talks and negotiations, and it was unclear what there was for the two countries - still technically at war - would have to discuss if not a deal to address concerns about the North's nuclear weapons program.
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