Following his arrival in Seoul from Pyongyang on Sunday, Pompeo said at a meeting with South Korean President Moon Jae-in that he had "a good productive conversation" with Kim, in talks which represented "another step forward". "We'll continue to make significant progress and we are further along in making that progress than any administration in an awfully long time".
The US official said earlier on Monday that "great progress" had been made with Kim Jong-un, and that work is being done to schedule a second summit with US President Donald Trump.
Speaking to reporters en route to Japan, Pompeo said the plan was to "develop options for both location and timing" for a second summit, although he doubted they would "get it nailed" down mostly due to scheduling and logistics issues.
Trump said on Twitter that, "Progress made on Singapore Summit Agreements!" "We are not quite there yet". "But we do think right that this is a place where ultimately some of these big hard issues have to be resolved by the nation's most senior leaders and hope to have those presented in a way that the two leaders can resolve them when they get together".
North Korean state media KCNA said: "Kim Jong-un expressed satisfaction over the productive and wonderful talks with Mike Pompeo at which mutual stands were fully understood and opinions exchanged". Korean Central News Agency (KCNA) reported that Kim had "warmly welcomed" Pompeo in Pyongyang and "spoke highly of him".
The U.S. Secretary of State is making one last stop before returning home.
"We require that the U.S. stop such misguided actions", he pointed out, adding that the two countries should pursue cooperation "and not descend into conflict and confrontation".
Valentina Matviyenko, chairwoman of Russia's Federation Council, revealed on Friday that Moscow and Pyongyang were working to arrange a visit for Kim, while Beijing was reportedly weighing options for a fourth meeting between Xi and Kim.
"It's a very nice day that promises a good future", Kim said, "for both countries".
"I don't have much to add", Mr Pompeo replied.
The latest chapter in the diplomatic flurry comes as Trump sent Pompeo on a whirlwind trip to the region to rekindle stalled denuclearization talks with Pyongyang.
Pompeo has repeatedly refused to discuss details of negotiations, including a USA position on North Korea's demand for a declared end to the Korean War and a proposal from Seoul for such a declaration to be accompanied by a shutdown of the North's main known nuclear facility.
North Korea's official name is the Democratic People's Republic of Korea or DPRK.
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Pompeo, however, played down the differences.
"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.
Pompeo said this could happen "as soon as we get it logistically worked out", adding: "Chairman Kim said he's ready to allow them to come in".
Trump and Kim held an historic first summit in Singapore in June.
He said North Korean leader Kim Jong-un had agreed to allow global inspectors into the Punggye-ri nuclear test site that was dismantled in May. It has not taken any steps to halt nuclear weapons or missile development. A series of diplomatic understandings on Taiwan - in which the Communist Party tolerates American weapons sales while the us accepts Beijing as the capital of "one China" - underpin more than four decades of relations between the two sides.
Moon also said there was a possibility of Kim holding a summit with Japanese prime minister Shinzo Abe.