Mr Kim's sister Kim Yo Jong delivered the invitation during a meeting at the presidential palace in Seoul
North Korean leader Kim Jong Un has invited South Korean president Moon Jae-in to a summit in Pyongyang.
According to a spokesperson for Mr Moon, Mr Kim's sister Kim Yo Jong delivered the offer on her brother's behalf while at the presidential palace in Seoul.
The North Korean leader is reported to be willing to meet his southern counterpart 'at the earliest opportunity'.
Kim Yo Jong was the most high-profile guests for the opening ceremony of the Winter Olympics in Pyeongchang, sharing an historic handshake with Mr Moon at the ceremony.
The Winter Olympics have seen officials from both countries take some tentative steps towards reopening negotiations and diplomatic relations.
Both countries marched under the same banner at the opening ceremony, while Ms Kim led the most senior delegation from the north to visit its neighbour since the Korean War.
It culminated in Ms Kim meeting Mr Moon at the presidential palace, during which the handwritten invitation to the North Korean capital was delivered.
Ms Kim also left her own note during the visit, expressing hope for closer relations between the two nations:
Kim Yo Jong’s note when she visited the South Korean Presidential Office. “I hope Pyongyang and Seoul will become closer in the hearts of Koreans and will bring unification and prosperity in the near future.” pic.twitter.com/BsAtD3J2w4— Anna Fifield (@annafifield) February 10, 2018
There remains significant obstacles to any efforts to improve diplomatic relations between the two countries, including the tense relationship between the US - a key South Korean ally - and North Korea.
US vice president Mike Pence was among the other dignitaries at the opening ceremony, but he did not interact with Ms Kim despite sitting only metres away from her.
In South Korea, the prospect of reunification remains divisive - with support across the population dropping from 68.3% to 57.8% in the last for years.
Opposition is particularly strong from younger people, with a poll by the Korea Institute for National Unification last year indicating that more than 70% of South Koreans in their 20s opposing reunification.