A United States official added that the visit was "better than the last time" but further efforts were needed.
Pompeo visited Tokyo on Saturday and was expected in Seoul later on Sunday.
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".
"Well, I am really pleased for this opportunity".
Trump and Kim met in Singapore in June, and that summit ended with Kim's pledge to denuclearize the Korean Peninsula.
Kim said he believed the dialogue between the two countries would continue to develop favorably "based on the deep confidence between the two leaders of the two countries", and expressed gratitude to Trump for making a honest effort to implement the agreement made at their historic June summit, KCNA said.
"Thank you for hosting, President Trump sends his regards".
In a statement, a State Department spokesperson said Kim had invited inspectors to visit the Punggye-ri nuclear test site to confirm it has been irreversibly dismantled, without identifying which inspectors or further details.
The trip was the latest turn in a diplomatic saga that saw Trump and Kim threaten each other with nuclear war previous year only for them to hold their historic summit in Singapore.
"He wrote me attractive letters, and they're great letters", Trump said.
The visit was Pompeo's fourth to the North Korean capital.
On his way to Asia from Washington, Pompeo stuck to his guns, indicating that the spirit of the agreement between Trump and Kim at Singapore in June was that "we will get to denuclearization in a fully verified, irreversible way, and then we will actually deliver on the commitments to make this brighter future for the North Korean people". South Korea is urging the U.S.to make its next demand the dismantlement of North Korea's main nuclear facility, Yongbyon, instead of declaring its inventory, which could lead to squabbling without trust-building actions, according to Kang.
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-North Korea summit "at the earliest possible date" and continue to discuss the specifics of date and location, Moon's chief press secretary, Yoon Young-chan, said in a written release.
Trump commented quickly on Twitter that Pompeo and Kim "had a good meeting" where they discussed agreements reached during the first summit, in Singapore.
KCNA also said the two sides agreed to hold working negotiations for the second summit as early as possible. North Korea had committed to this previously but has been resistant to allowing worldwide inspectors to visit sites it has claimed have been dismantled.
"South Korea seems overly eager to improve relations and offer benefits to North Korea without requiring commensurate progress in reducing the North Korean security threat", said Klinger.
That is not how Pyongyang sees things, nor is it the approach being advocated by Seoul.
With talks largely deadlocked, at the heart of this round of negotiations is a debate over what the USA should demand from North Korea and what it is willing to give in exchange.
In an interview with the Washington Post, she said the North could agree to dismantle Yongbyon, its signature nuclear site.
The IAEA said in August it did not find any indication that North Korea had stopped its nuclear activities, adding to doubts about the country's willingness to abandon its arsenal.
Nauert also could not name who Pompeo would be meeting with in Beijing - just that "there's the expectation that he meets with someone who's his appropriate counterpart".
While North Korea blew up the entrances of the tunnels in May and removed the diagnostic equipment, the shafts could be reopened or new tunnels could be excavated if Kim chose to restart underground test blasts, he said.
Moon has been enthusiastic about a peace process with North Korea: Last month, the two Korean leaders held their third-ever meeting, the first meeting to take place outside the demilitarized zone.