Pompeo said "he agreed with Chairman Kim to hold the second US-North Korea summit at the earliest date possible", South Korea's presidential office said in a statement, although no specific time or location has yet been agreed.
Trump met Kim in Singapore in June for the first-ever summit between the two countries, resulting in what critics say was only a vague commitment by Kim towards denuclearisation of the Korean peninsula. It came amid expectations that his meeting with leader Kim could provide momentum to get the denuclearization process going again after a monthslong stalemate over how to rid the North of its nuclear weapons program. He said it set up conditions for another Trump-Kim summit and help denuclearisation of the Korean peninsula. 'We said, "Point number one: denuclearization".
The Secretary of State added that South Korea has been an integral part of negotiations aimed at denuclearising North Korea, and he passed along Mr Trump's gratitude for the country's efforts.
Stephen Biegun, the USA special representative for North Korea who accompanied Pompeo to Pyongyang, said he offered to meet his counterpart, Vice Foreign Minister Choe Son Hui, "as soon as possible" and they were in discussion over time and place.
During the last three days of his Asia trip, Pompeo met with top leaders from all three nations in Japan, South and North Korea.
US Department of State spokeswoman Heather Nauert seconded the "unified" approach toward the North, but cited "areas where the United States and China do not agree, including on the South China Sea and human rights".
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Pompeo said there are many issues on which Washington and Beijing disagree but it is important that both sides listen to each other. We continue to make progress on agreements made at Singapore Summit.
The North's regime since then has allowed Catholic organisations to run aid projects in the impoverished country, but direct relations with the Vatican are non-existent.
Wang Yi said he told Pompeo during a meeting Monday that a shadow has been cast "over the future of China-U.S. relations".
"North Korea is bolstering its ties with China and Russian Federation so although the U.S. is maintaining the sanctions regime, it's on thin ice", said Hong Hyun-ik, an analyst at the Sejong Institute.
Pompeo declined to comment whether there was progress on a shutdown of the Yongbyon site.
Two of the five people on the American side of the meeting table dealt with North Korea - new special representative for North Korea Stephen Biegun and National Security Council adviser Allison Hooker - reflecting Pompeo's desired focus. A second summit in the coming weeks could boost Trump ahead of US midterm elections in November, but a failure to secure any measurable commitments would undercut what the president has seen so far as a clear foreign policy strength. Trump hinted of more progress during a news conference in NY last month, and Pompeo said Monday that he would only discuss details the two sides had agreed to release. "After having a nice meeting we can enjoy a meal together", Kim told Pompeo as they walked down a hallway toward a guesthouse dining room.