"When President Moon asked the pope if it would be alright for Kim to send an invitation... the pope said: "What you just told me is enough, but it would be good if a formal invitation is sent", he said, speaking in Korean.
The South Korean leader signed a broad agreement with Kim last month created to reduce military tensions on the peninsula.
Moon expressed his gratitude to the pontiff and the Vatican for praying for peace on the divided peninsula half a world away.
"The pope told President Moon that he would immediately respond to the invitation once he officially receives one", Yoon continued. The Vatican insisted at the time a papal visit would only be possible if Catholic priests were accepted in North Korea.
Taiwanese Vice President Chen Chien-jen made the invitation during a weekend audience with the pope, later telling reporters that Francis had "smiled" when he made the offer.
But both sides will set up guard posts at the entry points to prevent defections, and for now North and South Korean civilians will visit the JSA at different times to prevent them running into each other.
Francis, however, has taken a less-absolutist approach and recently agreed to a deal with China over bishop nominations signed with China, which only allows religious practice in state-sanctioned churches.
The Vatican has no official diplomatic relations with North Korea. The North is estimated to have around 2,500 Catholics, compared to more than 5.5 million in South Korea.
"It will be a great help to my people who wish for peace".
The Vatican secretary of state, Cardinal Pietro Parolin, said of the likelihood of Francis accepting such an invite: "I would say the availability to go is there". In a previous interview with CNN, President Moon said he was against any deployment of nuclear weapons in South Korea, and hoped denuclearization could be achieved within a "similar Catholic framework" with the pope's assistance.
It spoke only of "the promotion of dialogue and reconciliation between Koreans" and "the common commitment to fostering all useful initiatives to overcome the tensions that still exist in the Korean Peninsula, in order to usher in a new season of peace and development". Kim also held an unprecedented summit with U.S. President Donald Trump in Singapore in June, and promised to work toward denuclearisation of the Korean peninsula.