Pompeo arrived in China from North Korea and said he wanted to inform Yi about his meeting with the North Korean leader.
Pompeo on Monday hailed "significant progress" in weekend talks with Kim, and said the two sides were "pretty close" to agreeing on details for a second summit between Trump and Kim to follow their unprecedented June 12 meeting.
"A new order is being created on the Korean Peninsula", Moon declared. They said the North Korean leader appeared simply to be repackaging and dragging out past pledges.
"I like him, he likes me, the relationship is good", Trump said of Kim.
Pompeo's retort came after Wang accused the United States of escalating trade disputes, interfering on Taiwan and meddling in the country's domestic affairs. US officials have thus far said the penalties will remain in place until the North's denuclearization is fully verified.
Since Trump and Kim met in Singapore in June, US-North Korea relations have stalled.
While Moon, Kim, and members of both sides made casual conversation at the symbolic mountain's peak on September 20, the Archbishop said he would convey to Pope Francis that inter-Korean relations were "improving and moving in the direction of peace", Kim said during Tuesday's briefing.
Kim told Moon of his wish to meet the pope during last month's summit of the two leaders, the spokesperson added.
The North Korean dictator has invited the Pope to meet in Pyongyang.
The United States has also angered China with arms sales to Taiwan and new rules allowing top-level U.S. officials to travel to the island, though Washington still recognises Beijing over Taipei.
"As soon as we get it logistically worked out, Chairman Kim said he's ready" to "allow them to come in", and once the arrangements are made "we'll put them on the ground", Pompeo said. Pompeo was in Tokyo for talks with Japanese officials ahead of his trip to North Korea.
Moon said Kim was expected to visit Russian Federation soon and that Chinese leader Xi Jinping was also expected to travel to North Korea, but did not elaborate. "We've made incredible progress - beyond incredible".
His comments came at the start of a meeting with Pompeo at the Chinese capital's Diaoyutai Guest House.
Pompeo shot back that the U.S.is very concerned with China's risky actions and rhetoric, which destabilizing for cross-strait relations and the entire region. The North has rejected that and South Korea has suggested it may not be a necessary next step.
South Korea's presidential office said in a statement that Kim told President Moon Jae-in during their summit last month that the pope would be "enthusiastically" welcomed in North Korea.
The Beijing stop could be tense as it comes days after Vice President Mike Pence delivered a blistering speech accusing China of military aggression, commercial theft, rising human rights violations and electoral intervention against Trump.
In Seoul, before leaving for talks with Chinese officials in Beijing, Pompeo said Kim is expected soon to name deputy foreign minister Choe Son Hui as a counterpart for his new special envoy for North Korea, former Ford executive Stephen Biegun, who accompanied him on the trip.
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