Kim and Pompeo agreed to arrange a summit between the North Korean leader and U.S. President Donald Trump "as soon as possible", and discussed potential U.S. monitoring of Pyongyang's steps toward denuclearization, South Korea's presidential office said on Sunday.
In Pyongyang, Pompeo and Kim met for almost four hours, first in a business session and then in a 90-minute luncheon that the North Korean leader hosted at a state guest house.
He would not be specific but said he and Kim had agreed to shortly begin working-level talks on the nuts and bolts of denuclearization, on the placement of global inspectors at one of North Korea's main nuclear facilities, and had come close to finalizing a date and venue for the next Kim-Trump summit.
The South Koreans said the US and North Korea discussed establishing formal negotiating groups to set a "denuclearization process" and work out a time and location for a follow-up meeting to the first Trump-Kim summit in June.
The United States wants sanctions on North Korea to remain in place until the country completely dismantles its nuclear program.
As they sat for lunch, Kim said, "It's a very nice day that promises a good future for both countries". Pompeo did not meet Kim on that trip.
Instead, the top US diplomat has been tasked by his boss with preparing for a second summit with Kim Jong Un, which the North Koreans are pushing for, but that analysts warn will hurt USA goals.
But that could end if the US agrees to sign a declaration to officially end the Korean War, a major demand on the North Korean side and something Pompeo has been open to.
Kim said the bilateral dialogue would continue to develop "based on the deep confidence between the two leaders", and expressed gratitude to Trump for making a honest effort to implement the agreement made at their June summit, KCNA said.
Pompeo said he and Kim discussed denuclearisation steps to be taken by the North and the issue of US government monitoring of those actions, which Washington sees as vital, as well as the measures the United States would conduct in return, Moon's office said.
Nauert said Pompeo "held productive discussions" Kim on Sunday.
Pompeo is heading into his eighth encounter with the North Korean side - and his fourth meeting since the Singapore summit.
"Yeah, so we had a great, great visit this morning", Pompeo replied. Pompeo arrived in Tokyo for talks with Japanese officials ahead of his trip to North Korea. "I thought we had a good, productive conversation, and as President Trump has said, there are many steps along the way, and we took one of them today, another step forward".
While North Korea blew up the entrances of the tunnels in May and removed the diagnostic equipment, the shafts could be reopened or new tunnels could be excavated if Kim made a decision to restart underground test blasts, he said.
In that same series of tweets, Trump also linked the stall in denuclearization to U.S.
Later Sunday, Pompeo flew to Seoul where he met South Korean President Moon Jae-in.
South Korean President Moon Jae-in (R) shakes hands with United States Secretary of State Mike Pompeo during their meeting at the presidential office in Seoul on October 7, 2018.
Both Koreas have differed with the United States on North Korea's nuclear program. South Korea is urging the U.S.to make its next demand the dismantlement of North Korea's main nuclear facility, Yongbyon, instead of declaring its inventory, which could lead to squabbling without trust-building actions, according to Kang.
Another planned trip by Pompeo to Pyongyang was scrapped after what Trump said was insufficient progress towards implementing the terms of the Singapore declaration.
In Tokyo on Saturday, Pompeo promised he would bring up Japan's concerns about North Korea's alleged abductions of Japanese citizens and its nuclear ambitions.
Pompeo will also visit Beijing Monday as he concludes his four-nation visit to the region.