For his part, Moon said that "I dearly hope that your latest visit, as well as the upcoming U.S".
Shortly after arriving in South Korea, Pompeo posted a photo of himself walking along with Kim on Twitter, saying: "Had a good trip to #Pyongyang to meet with Chairman Kim".
"I am really pleased for this opportunity".
In a more upbeat note, Pyongyang's state media said on Monday Kim lauded his talks with Pompeo, where Kim "explained in detail the proposals for solving the denuclearization issue".
He also included three photos of Pompeo and Kim together.
Moon said a second Trump-Kim summit could be accompanied by major diplomatic developments that could contribute to the denuclearization of the Korean Peninsula and stabilizing peace.
Pompeo said he had a "great visit" and a "very successful morning", adding that Trump sent his regards.
The North Korean leader also invited inspectors to visit the nuclear test site that was destroyed back in May to "confirm that it has been irreversibly dismantled". Pompeo did not meet Kim on that trip.
Pompeo responded: "We had a great, great visit this morning".
In recent weeks, US officials have pointed to a recent meeting in Pyongyang between Kim and Moon, when North Korea committed to dismantling a missile-engine-testing site and raised the possibility of doing the same at its Yongbyon nuclear-enrichment facility.
"Most importantly, both the leaders believe there's real progress that can be made, substantive progress that can be made at the next summit", Pompeo said.
After a previous visit to Pyongyang in July, Pompeo had said the two foes made progress on key issues - but within hours of his departure the North condemned "gangster-like" demands from the U.S., raising questions over how much the two sides really saw eye to eye.
Napoli vs. Liverpool - Football Match Report
Gini Wijnaldum and James Milner start in midfield, whilst Keita replaces Jordan Henderson in the starting eleven. The Champions League clash against Napoli represents his chance to show Klopp just how good he can be.
It added that the two sides agreed on holding working negotiations for the second North Korean-U.S. summit as early as possible. "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".
Pompeo has insisted that sanctions on North Korea should remain in place until the country completely dismantles its nuclear program.
Another planned trip by Pompeo to Pyongyang was scrapped after what Trump said was insufficient progress towards implementing the terms of the Singapore declaration.
Pompeo was joined by USA special representative on DPRK issues Stephen Biegun, and Kim was joined by Kim Yo Jong, first vice department director of the Central Committee of the ruling Worker's Party of Korea (WPK), the statement said.
"South Korea seems overly eager to improve relations and offer benefits to North Korea without requiring commensurate progress in reducing the North Korean security threat", said Klinger.
Moon had his own third summit with Kim last month in Pyongyang, which was partly meant to help salvage the stumbling negotiations between Pyongyang and Washington, after Trump called off Pompeo's planned visit to the North in late August citing lack of progress.
That is not how Pyongyang sees things, nor is it the approach being advocated by Seoul.
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.
Analysts say Washington may now consider new options as China, Russia and South Korea seek to relax sanctions.
Absent sanctions relief, North Korea has rejected turning over what Washington really wants in return: a complete inventory of its nuclear and ballistic missile facilities that could be used by worldwide inspectors to verify they have been dismantled.
Nauert also could not name who Pompeo would be meeting with in Beijing - just that "there's the expectation that he meets with someone who's his appropriate counterpart".