"I support the courageous decisions of the North and South Korean leaders", the pope said following the first summit in Panmunjom in April.
"China and U.S. relations are at an important juncture and facing challenges", Yang said.
Dmitry Peskov, spokesman for Russian President Vladimir Putin, said Monday that Putin extended an invitation to Kim last month to visit Russia, and that Moscow is working on the details of where and when that meeting could happen. Also, North Korean Vice Foreign Minister Choe Son-hui visited Beijing and Moscow last week and reportedly discussed arrangements for Kim's visits.
Pompeo said there are many issues on which Washington and Beijing disagree but it is important that both sides listen to each other.
Pompeo met with the North Korean leader in Pyongyang on Sunday to rekindle stalled denuclearisation talks following a landmark summit between Kim and US President Donald Trump in Singapore.
The trade war between China and the United States is however being seen as a barrier to co-operation in the North Korean crisis.
He earlier said "significant progress" was made during his visit toward an agreement for North Korea to give up its nuclear weapons.
The US should cease military contacts and halt further arms sales to Taiwan, and it should stop interfering with Chinese efforts to establish diplomatic relations with Taiwan's diplomatic allies, he said.
Meanwhile, in Washington, foreign relations experts noted that with 90 percent of North Korea's exports going through China, Beijing is the one calling the shots.
"Chairman Kim was very clear yesterday in our conversation about the things that we need to do between the United States and the DPRK", Pompeo described.
Things didn't go much better a day later in Beijing, where Pompeo had the bad fortune of being the first senior USA official to visit the country since a blistering speech on China last week from Pence. "An announcement of a near-term summit would show that the Trump administration has again prioritized a flashy show over real substance", he said. Pompeo did not meet Kim on that trip. USA officials have thus far said the penalties will remain in place until the North's denuclearization is fully verified.
Moon also said the North will "permanently dismantle" its missile engine testing site and launch platform in the northwestern town of Tongchang-ri in the presence of experts from "concerned countries".
Kim also expressed the belief that - "based on the deep confidence between the top leaders of the two countries" - dialogue and negotiations would continue to develop favorably and a "good program" for a second summit would be developed. "But if you look at the approach, we are in lockstep with each of those two countries in terms of how we approach achieving the results that everyone is aimed at".
The Vatican had no immediate comment on the North Korean leader's invite.