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.
South Korean President Moon Jae-in said that Kim is expected to visit Russian Federation soon and that Chinese leader Xi Jinping will also travel to North Korea, but did not give any further details.
Pompeo said he and Kim had gotten "pretty close" to fixing the logistics for the summit but stressed that "sometimes that last inch is hard to close".
In response, Kim had "expressed his will and conviction that a great progress would surely be made in solving the issues of utmost concern of the world and in attaining the goal set forth at the last talks", North Korea's state-run news agency KCNA reported on Monday, adding the talks were "productive and wonderful".
"We are starting to see a first wave of actions we can take on all four pillars of the Singapore communique", said Biegun, who will serve as Pompeo's pointman in negotiations with the North as well as working with South Korea, Japan, China, Russia and other countries that have an interest in the talks.
Pompeo landed in Pyongyang on Sunday morning to meet Kim after visiting Japanese Prime Minister Shinzo Abe on Saturday in Tokyo.
In a statement, State Department spokeswoman Heather Nauert said Kim had invited inspectors to visit the Punggye-Ri test site in order to confirm that it had been dismantled.
"It's a very nice day that promises a good future for both countries", Kim told Pompeo through an interpreter as they sat down for the meal.
But Pyongyang failed to keep its pledge to allow global inspections of its demolition of the Punggye-ri site in May, fanning criticism that the move could be reversed.
DPRK is the acronym for the North's official name, the Democratic People's Republic of Korea.
Pompeo, who arrived from North Korea, told Yi he wanted to fill him on his meetings with its leader, Kim Jong Un, and efforts to achieve the denuclearization of the Korean Peninsula. He lamented Beijing's recent decision not to attend the strategic dialogue planned between the two countries' defense secretaries, saying that the forum was an "important opportunity" for discussion.
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.
He said, "Thanks for hosting me and my team".
In an interview with the Washington Post, she said the North could agree to dismantle Yongbyon, its signature nuclear site.
"Most importantly, both the leaders believe there's real progress that can be made, substantive progress that can be made at the next summit", Mr Pompeo said.
Xi has not visited North Korea since coming to power in 2013 although he sent Li Zhanshu, Beijing's third-ranking Communist Party official, to Pyongyang last month for North Korea's 70th anniversary celebrations.
"We can't rule out the possibility that Washington. may move in the direction of partial easing of sanctions based on progress in denuclearisation", he said. "I thought we had a good, productive conversation, and as President Trump has said, there are many steps along the way, and we took one of them today, another step forward".
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