He cited North Korean leader Kim Jong Un's decision to "complete the previously announced dismantlement of the Tongchang-ri site in the presence of USA and global inspectors as a step towards the final fully verified denuclearisation of North Korea".
Kim, who recently proposed another summit with Trump after their unprecedented June talks in Singapore, said the North was willing to "permanently dismantle" key missile facilities in the presence of outside experts, and the Yongbyon main nuclear complex, if the United States took corresponding action.
Among possible Pyongyang demands are the permanent end to "provocative" military exercises between the US and South Korea, which North Korea says amount to invasion practice.
Moon said Kim wanted Pompeo to come to North Korea as well.
South Korean President Moon Jae-in, second from right, and his wife Kim Jung-sook, right, stand with North Korean leader Kim Jong Un, second from left, and his wife Ri Sol Ju on the Mount Paektu in North Korea.
Moon said Kim had also told him the closure of a nuclear weapons test site at Punggye-ri could be subject to worldwide verification.
Washington has demanded concrete action, such as a full disclosure of North Korea's nuclear and missile facilities, before agreeing to Pyongyang's key goals, including an easing of global sanctions and an official end to the Korean War.
Moon's office said North Korean workers made a mistake.
According to Pyongyang's narrative, it is also where Kim Jong Un's father Kim Jong Il was born, at a secret camp where his own father Kim Il Sung, the North's founder, was fighting Korea's Japanese colonial occupiers.
"We absolutely can not let this hard to come by opportunity for peace slip away once again", the Chinese government's top diplomat, State Councillor Wang Yi, said in a statement.
Eliud Kipchoge breaks world marathon record in Berlin
I also congratulate his compatriots Amos Kipruto & Wilson Kipsang for going out valiantly to bring a 1-2-3 victory for #TeamKenya. By 40 kilometres, reached in 1:55:32, a world record was a certainty. "The greatest marathon runner of all times", they said.
"He's calm, I'm calm - so we'll see what happens", Trump, who past year threatened to destroy North Korea, told reporters.
U.S. officials said the ambiguity about what Washington was supposed to do for the North to close its nuclear complex at Yongbyon gave Kim room to argue that Washington had not done enough for North Korea to follow through on its pledges.
Ahead of this week's talks, it was expected that two leaders would continue to work to formally end to the Korean War, which ended in an armistice 65 years ago.
Other than that, however, there are scant details about what Pyongyang's demands are, or what concrete steps it would take to fulfill Kim's commitment to Trump in June. "Prior to becoming president, it looked like we were going to war with North Korea, and now we have a lot of progress", Mr. Trump said.
Moon said he and Kim spent most of a three-day summit discussing how to break an impasse and restart nuclear talks between Pyongyang and Washington.
The games marked the end of the South Korean president's second day in the capital.
The leaders are basking in the glow of the joint statement they signed Wednesday.
Kim Jong Un, too, has visited the mountain around major developments in North Korea, such as before he executed top officials including his uncle Jang Song-thaek in late 2013, and following North Korea's fifth nuclear test in 2016. They also agreed to withdraw 11 guard posts from the Demilitarized Zone by December and to establish a no-fly zone above the military demarcation line that bisects the two Koreas that will apply to planes, helicopters and drones.
On Wednesday, Kim also said he plans to visit Seoul "in the near future" - a trip that would be historic in the relationship of these two countries that technically remain at war.