The message will be formally delivered to the Pope by President Moon of South Korea when he visits the Vatican this month.
"For Asia, the severity of China-U.S. frictions is taking up much attention and is, to some extent, diluting attention paid to the Korean peninsula issue", it said.
"Chairman Kim (Jong-un) said he will "ardently welcome the Pope if he visits Pyongyang"," said presidential spokesperson Kim Eui-kyeom, according to Yonhap news agency.
"President Moon had said the Pyongyang summit must receive the support of the global community, and suggested [inviting the pope] would be one way" of earning trust, a South Korean presidential Blue House source told Hankyoreh.
Pompeo responded by saying that Washington and Beijing were stuck in a "fundamental disagreement" that he hoped they could make some progress on resolving.
"There was an exchange of stands of the top leaderships of the two countries on the issues arising in holding the second DPRK-US summit talks to be followed by an in-depth discussion of them", the KCNA said.
Kim also brought up the interactions between the North Korean leader and Roman Catholic Archbishop Hyginus Kim Hee-joong at Mt. Paektu last month, who was invited to join the South Korean delegation alongside President Moon.
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. In June, NBC News reported that an updated intelligence assessment concluded that North Korea was stepping up production of fuel for nuclear weapons at secret sites across the country.
"North Korea is bolstering its ties with China and Russian Federation so although the U.S. is maintaining the sanctions regime, it's on thin ice", said Hong Hyun-ik, an analyst at the Sejong Institute. A second summit in the coming weeks could boost Trump ahead of US midterm elections in November, but a failure to secure any measurable commitments would undercut what the president has seen so far as a clear foreign policy strength.
North Korea took apart the Punggye-ri facility in the country's northeast in May but has yet to allow worldwide observers into the site to verify its claims.
Trump "will not back down" from the challenge, Pence said.
Later Kim said, "It's a very nice day that promises a good future" for both countries.
In a statement, State Department spokeswoman Heather Nauert said Kim had invited inspectors to visit the Punggye-Ri test site in order to confirm that it had been dismantled. Pompeo did not meet Kim on that trip.
The two countries have, however, been in a standoff over what the term denuclearisation means.
The Vatican did not comment on the possibility of a papal visit.
During Mr Pompeo's last visit to Pyongyang in July, he was denounced for making "gangster-like demands".
It is reportedly the same plane that transported Kim's limousine and personal effects to Singapore for his meeting with President Trump.