Chinese leader Xi Jinping is also expected to travel to North Korea.
"North Korea summit, Chairman Kim Jong Un's visit to Russian Federation and President Xi Jinping's visit to North Korea are expected to take place soon", Moon told a cabinet meeting. Korean Central News Agency (KCNA) reported that Kim had "warmly welcomed" Pompeo in Pyongyang and "spoke highly of him".
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.
President Trump and Chairman Kim held an historic summit on June 12.
"Kim Jong Un expressed satisfaction over the productive and wonderful talks with Mike Pompeo at which mutual stands were fully understood and opinions exchanged", KCNA said.
Kim praised their "nice meeting", telling Pompeo via an interpreter following the morning's talks that it was "a very nice day that promises a good future. for both countries".
In Seoul, Pompeo told South Korean President Moon Jae-in that Kim had agreed to have another summit "as soon as possible", according to a statement from Moon's office.
A report in 2014 by the United Nations Human Rights Council concluded that hundreds of South Koreans, Japanese and other foreign nationals were kidnapped by North Korea after the end of the Korean War in 1953.
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Pompeo also invoked the cancellation of a meeting between US Defense Secretary Jim Mattis and his Chinese counterpart, which was supposed to take place this month in Beijing.
Another planned trip by Pompeo to Pyongyang was scrapped after what Trump said was insufficient progress towards implementing the terms of the Singapore declaration.
Speaking to the press, Pompeo said that inspectors would be allowed at the site "as soon as we get it logistically worked out".
The North Korean leader has yet to meet with Russian President Vladimir Putin, who has previously invited Kim to Moscow.
"The United States is continuing to act against Chinese interests, including the U.S".
If ties between the two countries continued to deteriorate, there could be "profound changes" in the strategic environment for such regional issues as North Korea, China's state-backed Global Times tabloid warned in an editorial.
On Sunday, Pompeo met Kim during a trip to Pyongyang, his fourth visit to the country this year.
The US diplomat also met with Xi during a similar trip to the region in June.
"I know that during your visit, you also want to exchange views with the Chinese side on regional hotspot issues including the situation in the Korean Peninsula", Wang said.
Washington and Pyongyang have sparred over the exact terms of the vaguely-worded agreement in Singapore, with the USA pushing to maintain sanctions and pressure against the North until its "final, fully verified denuclearisation".