Kim has been intensely engaged in diplomacy in recent months in what's seen as an effort to leverage his nuclear weapons program for an easing of economic sanctions and military pressure.
South Korea had shut down water and power supplies as it closed the Kaesong factory park in February 2016 following a North Korean nuclear test and long-range rocket launch.
Moon plans to convey Kim's desire for a papal visit when he travels to the Vatican next week.
The two leaders exchanged messages amid expectations that Kim will visit Russian Federation to hold talks with Putin in the near future.
She later backtracked, saying she had misspoken, and her ministry said no active review was in place. But on this, his fourth visit to Pyongyang, he and Kim had lunch together before Pompeo left for South Korea.
The two Koreas have held three summits this year. "The suffering of the division is great, and I understand this and pray that it ends".
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"In a situation where there is ongoing dialogue on denuclearization and the advancing of inter-Korean relations, there is need for flexibility in such a review, without infringing upon UN Security Council resolutions and sanctions on the North". Since Trump and Kim met in Singapore, relations between their nations have stalled and questions have emerged about the accord they signed, which had no timetable and no measurable factors to show North Korea was denuclearizing.
South Koreans are deeply divided along ideological lines and many people still harbor deep anger over North Korea's 1950 attack that started the Korean War, killing and injuring millions.
Amid an impasse in denuclearization talks, Washington and Pyongyang could strike a deal whereby the threat to the US mainland is removed but the North keeps its nukes, and they would certainly be capable of hitting anywhere in South Korea.
South Korean government ministries are continuing to review the possibility of lifting the country's so-called "May 24" sanctions against North Korea, ROK foreign minister Kang Kyung-wha said on Wednesday.
The Vatican's priests were expelled by North Korea long ago and state-appointed laymen officiate services.
This is all the more important because North Korea's rights issues are now being overshadowed by denuclearisation and other security issues.