"During the meeting with Pope Francis, he (Moon) will relay the message from chairman Kim Jong Un that he would ardently welcome the pope if he visits Pyongyang", Moon's spokesman, Kim Eui-kyeom, told reporters.
The unprecedented summit served as a platform for the two leaders to discuss denuclearising the Korean peninsula.
But Kim Il Sung, the North's late founding leader and the current ruler's grandfather, viewed Christianity as a threat to his authoritarian rule and eradicated it through executions and labor camps.
Meanwhile, planning continues for a second summit between Mr Kim and US President Donald Trump. But that wasn't an advance on comments Trump made last month in NY, when he said he wanted to meet Kim "very soon".
High-level exchanges between the two countries have been frozen, including a planned diplomatic and security meeting between US Secretary of Defence James Mattis and Chinese defence minister Wei Fenghe originally scheduled for mid-October in Beijing. Daniel Russel, the chief USA diplomat for East Asia until past year, said on NBC.
While Trump is seeking a second meeting with Kim - possibly ahead of the November 6 voting - he's sending Pence in his place to regional summits planned for next month in Singapore and Papua New Guinea.
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.
He said that Kim invited inspectors to the Punggye-ri test facility, the site of all six of the regime's nuclear blasts.
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.
Pompeo responded by saying that Washington and Beijing were stuck in a "fundamental disagreement" that he hoped they could make some progress on resolving.
Weakening support for USA -led sanctions - from countries including Russian Federation and China, but also South Korea - has also led to a growing sense that Pompeo is spending more and more of his time just trying to keep existing restrictions in place.
Pompeo, however, did not comment on possible corresponding measures to be taken by Washington.
There's no sign that North Korea is slowing down its production of nuclear fuel.
In South Korea, Pompeo appeared alongside President Moon Jae-in, who said the world wanted to know the results of the Pyongyang meeting, "so if the secretary can disclose the results, please tell us".
Kim "expressed his gratitude to President Trump for making honest effort" to implement the Singapore agreement, according a report on the Pompeo visit by North Korea's state news agency KCNA.
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.
"I will tell you they're calling, wanting to go there and wanting to invest", Trump said.