Al Jazeera's Rob McBride, reporting from the South Korean capital, Seoul, said Pompeo's meeting with Kim has "been taken as a positive sign that dialogue is still on track". "If there is a second meeting, they are pretty sure that they can get more out of Trump than just dealing with administrations officials, including Pompeo".
That idea has the support of some analysts who think the USA needs to contain the North Korea nuclear threat and defuse tensions overall.
He continued, "As President Trump has said, there are many steps along the way and we took one of those today; it was another step forward".
"What we all hope will be the denuclearization and the change in the relationship here on the peninsula... After having a nice meeting, we can enjoy a meal together".
He travels to Seoul Sunday evening to meet with President Moon Jae-in.
South Korean Foreign Minister Kang Kyung-wha told The Washington Post last week that Seoul believed that such a list could spark a long argument between Pyongyang and Washington over verification, which would not be conducive for building trust.
He did not meet Kim during that trip, which ended on poor terms.
The Koreas in past months have held a flurry of talks, including three summits, amid a global diplomatic push to resolve the nuclear standoff with North Korea.
"It's a very nice day that promises a good future. for both countries", he said, according to AFP.
The announcement Sunday came shortly after Pompeo arrived in Seoul, following his fourth visit to North Korea, where he met with Kim.
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During Pompeo's meeting in Pyongyang, Kim invited inspectors to visit the Punggye-ri nuclear test site to "confirm that it has been irreversibly dismantled", State Department spokeswoman Heather Nauert said in a statement. He is expected to leave for China on Tuesday.
Secretary of State Mike Pompeo visits the Foreign Ministry in Tokyo on Saturday before heading to Pyongyang. Pompeo smiled but said nothing in return.
While there is no way Pompeo, in roughly 210 minutes with Chairman Kim Jong Un, achieved any major breakthroughs Sunday, Pompeo may have accomplished his goal nonetheless: exploring the timing and possible locations for a second U.S.
"North Korea took some steps towards denuclearisation and the United States will face criticism from the global community if it continues to demand complete denuclearisation without any lifting of sanctions", said Yang Moo-jin, a professor at the University of North Korean Studies in Seoul.
Neither the USA nor the North Korean side released an immediate detailed readout on the content of Pompeo's talks with Kim.
Until recently, Pyongyang has appeared to be increasing its demands.
Abe called for "coordination" on this issue and also on North Korea's nuclear threat. "We are not even talking about whether to do it or not", he said. "It is such an oxymoronic, two-faced behavior'".
A USA official who was part of Mr Pompeo's delegation said that the trip was "better than the last time" but added: "It's going to be a long haul".
Pompeo also spoke about additional denuclearization steps and Washington's monitoring of the verification process and "corresponding measures", Yoon said.
"This is a busy time diplomatically", said Al Jazeera's McBride.