"Secretary Pompeo said he agreed with Kim to hold the second US-North Korea summit as soon as possible", Yoon Young-chan, the senior secretary to President Moon for public relations, said in a press briefing held after a meeting between Moon and Pompeo.
Prior to meeting with Moon, Pompeo tweeted that he "had a good trip to #Pyongyang to meet with Chairman Kim". Pompeo placed his hand on Kim's shoulder and they both smiled.
"I am really pleased for this opportunity".
Pompeo responded: "We had a great, great visit this morning". Both men spoke through translators.
Following meetings in South Korea, the US secretary of state will end his current tour with a stop in Beijing.
The North also has accused Washington of making "unilateral and gangster-like" demands on denuclearisation and insisted that sanctions should be lifted before any progress in nuclear talks. On that occasion, he did not meet Kim.
After Seoul, Pompeo ends his trip tomorrow in China, North Korea's political and economic lifeline.
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.
One U.S. official who accompanied Pompeo said Sunday's meeting was "better than the last time", but added that "it's going to be a long haul".
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.
"Like everything else Kim has done, it is a gesture that mimics disarmament that isn't actually disarmament", said Jeffrey Lewis director of the East Asia Nonproliferation Program at California's Middlebury Institute of International Studies.
The dovish Moon, who held three summits with Kim this year and also brokered the Trump-Kim summit in Singapore, said Sunday the "whole world" was watching with keen interest the outcomes of Pompeo's trip.
"North Korea summit, which I hope will be happening soon, will make an irreversible and decisive progress in terms of complete denuclearization on the Korean Peninsula as well as the peace process". -North Korea summit, which Trump predicted might be held "in the not too distant future".
Pompeo was also expected to discuss a declaration to end the Korean War on this trip.
There was no immediate indication whether Mr Pompeo had managed to arrange a much-anticipated second summit between Mr Kim and United States president Donald Trump.
In a more upbeat note, Pyongyang's state media said on Monday Kim lauded his talks with Pompeo. North Korea had committed to this previously but has been resistant to allowing global inspectors to visit sites it has claimed have been dismantled.
Analysts say Washington may now consider new options as China, Russia and South Korea seek to relax sanctions.
North and South Korea want a "phased" approach with Pyongyang rewarded as it it happens.
"I would like to have thorough coordination with you on our respective policies towards North Korea", Abe responded. It has not taken any steps to halt nuclear weapons or missile development.
Trump, tweeting from Washington, cited progress on agreements he made with Kim at their June meeting and said, "I look forward to seeing Chairman Kim again, in the near future".
Mr Pompeo had flown to Pyongyang from Tokyo after talks with Japan's prime minister Shinzo Abe during which he pledged the Trump administration would coordinate and unify its strategy for denuclearisation with allies.
After a summit between the leaders of the two Koreas last month, Kim said he was prepared to permanently dismantle his country's main nuclear site at Yongbyon, but only if the United States took "corresponding steps" to build trust.