Kim and Pompeo met for about two hours, and then had lunch together at the Paekhwawon, or 100 Flowers Garden, a prestigious state guesthouse, for another hour and a half, according to a pool report.
According to the South Korean statement, Pompeo said "he agreed with Chairman Kim to hold the second US-North Korea summit at the earliest date possible".
Pompeo was also aiming to set up a location and a time for a second summit between Trump and Kim, although he warned in advance that he might not be able to nail down all the details on this trip and probably wouldn't be in a position to announce anything right away. USA officials have thus far said sanctions will remain in place until the North's denuclearisation is fully verified. Trump canceled Pompeo's initial planned return to North Korea last month.
In Tokyo, Pompeo told Abe that the US wanted to work closely on nuclear talks and that he would raise with the North Koreans the issue of Japanese abductees held for decades by the regime.
North Korea's foreign minister said last week that Pyongyang will not disarm without first receiving concessions from Washington.
North Korean leader Kim Jong Un described his talks with U.S. Secretary of State Mike Pompeo on Sunday as "productive and wonderful", state media KCNA said on Monday, striking a positive tone compared with a previous state media stance when talks were stalled.
Moon said he hoped a Trump-Kim summit would happen soon, and would "make irreversible, decisive progress in terms of denuclearization as well as the peace process".
Moon also said there was a possibility of Kim holding a summit with Japanese Prime Minister Shinzo Abe. Pompeo said it is important for the two allies to have a "fully coordinated, unified view of how to proceed, which will be what is needed if we are going to be successful on denuclearizing North Korea". It is to be noted here that US President Donald Trump had met Kim in Singapore in June for the first-ever summit between the countries.
Pompeo replied: "So we had a great, great visit this morning". Pompeo arrived in Tokyo for talks with Japanese officials ahead of his trip to North Korea. President Trump sends his regards.
But several experts told ABC News that North Korea hadn't done enough yet to earn an end-of-war declaration, let alone a second Trump-Kim summit.
"North Korea took some steps towards denuclearisation and the U.S. will face criticism from the global community if it continues to demand complete denuclearisation without any lifting of sanctions", said Yang Moo-jin, a professor at the University of North Korean Studies in Seoul.
But the inter-Korean agreement again failed to stipulate any plans called for by the United States, such as to declare a list of its nuclear weapons, facilities and materials, or a concrete timeline for denuclearisation.
"It's a long process", Pompeo told a small group of reporters in the South Korean capital of Seoul where he traveled after meeting with Kim in Pyongyang on Sunday.
An official of the State Department was not immediately able to confirm Pompeo's arrival in Pyongyang, from where outside communication can be limited.
Trump has appeared eager to meet Kim again, even though the two sides are seen far from narrowing their differences. Instead, they want both sides to take a "phased" approach, in which Pyongyang is rewarded as it takes gradual steps to roll back its nuclear program.
Pompeo has insisted that sanctions on North Korea should remain in place until the country completely dismantles its nuclear program.
Also on Monday, South Korean President Moon Jae-in said he expects Kim to travel to Russian Federation and for Chinese President Xi Jinping to visit North Korea soon amid a global diplomatic push to resolve the nuclear crisis.