Mr Kim held an unprecedented summit with US President Donald Trump in Singapore in June, and pledged to work toward denuclearisation of the Korean peninsula.
South Korean President Moon Jae-in said on Monday he expected Kim to visit with both Putin and Chinese Communist Party leader Xi Jinping "soon", and a summit with Japanese Prime Minister Shinzo Abe is possible. "After having a nice meeting we can enjoy a meal together", Kim told Pompeo as they walked down a hallway toward a guesthouse dining room.
Pompeo did not say when inspectors would be allowed to Punggye-ri, and the State Department did not respond when asked if they would be Americans or others from worldwide nuclear bodies. "North Korean summit and we still don't even have a common definition of seemingly straightforward terms such as "denuclearization" and 'Korean Peninsula, '" Bruce Klingner, senior research fellow for Northeast Asia at the Heritage Foundation in Washington, said via email.
In exchange, the United States would declare a formal end to the 1950-53 Korean War - which concluded with an armistice rather than a peace treaty - but North Korea would stop short of delivering an exhaustive list of its nuclear facilities, she said.
"You have no nuclear tests, you have no rockets, and we have a very good relationship with Chairman Kim, which is very important", Trump said. DPRK is the acronym for the country's official name: Democratic People's Republic of Korea.
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That makes any focus on a second Trump-Kim summit misguided, Klingner of Heritage said.
"I returned late last night from North Korea from a trip where we made real progress".
The pontiff made a five-day trip to South Korea in August 2014, his first visit to Asia as pope, during which he called for peace and reconciliation on the Korean peninsula. The North Korean leader said progress is being made in implementing the goals of the first summit, and he "expressed his gratitude to President Trump for making honest effort to this end", KCNA said.
The North has accused Washington of making "unilateral and gangster-like" demands on denuclearization and has insisted that sanctions should be lifted before any progress in nuclear talks.
South Korean President Moon Jae-in is himself scheduled to meet the pontiff during his trip to Europe next week.
He is to work with South Korea, Japan, China, Russia and other countries that have an interest in the talks.
Pompeo declined to comment on the status of talks over Yongbyon or the details of negotiations generally, saying "we're not going to talk about where we are in these negotiations, except for things we have agreed to release with the North Koreans".