A formal invitation was delivered by Mr Kim’s sister, Kim Yo-jong during a rare meeting at the presidential palace in Seoul.
If the summit were to take place, it would be the first meaningful talks between the secretive regime and its southern neighbours for more than a decade.
Ms Kim, part of a delegation from the communist state visiting the South for the Winter Olympics, reportedly told Mr Moon he should visit the North at the “earliest date” possible.
The President said the two states should ensure the meeting could take place by “creating the necessary conditions in the future”.
“I wish I can see you in Pyongyang at an early date,” Ms Kim told Mr Moon, according South Korea’s state run Yonhap News Agency. She added: “If you meet Chairman Kim Jong-un and exchange views on many issues, the North-South relationship may quickly improve as if yesterday would seem a far distant past.
“I hope the President will put a footprint in history that will be long remembered by the future generation by playing a leading role in opening a new era of unification.”
Ms Kim is the first member of North Korea’s ruling family to visit the South since the Korean War ended in 1953 in a military stalemate and without a peace treaty.
The proposed summit would be only the third between the Koreas since the war ended, after meetings in Pyongyang in 2000 and 2007.
Mr Moon has seen the Winter Olympics in Pyeongchang as an opportunity to thaw relations between the North and the South, who technically remain at war.
Athletes from both countries marched together at the opening ceremony of the games under the flag of the Korean Peninsula. The President is also due to join North Korea’s ceremonial head of state, Kim Yong-nam, to watch a unified Korean women’s ice hockey team play their first match of the Olympics.
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