During Moon's visit to Pyongyang in September, Kim promised to pay a return visit to Seoul "at an earliest date", sparking speculation that he may visit by the end of this year.
Another speaker, in a reflection of the shift in perception towards Kim in South Korea, said: "Until now, we did not know Kim's true self".
"There will still be a lot of difficulties ahead", Moon said. "There's no change in our heart about welcoming Chairman Kim [to the South]".
The tweet also included a photo that showed a ruby-coloured folder emblazoned with the seal of Pyongyang's powerful State Affairs Commission and the top part of Kim's letter, which started with: "Dear your excellency President Moon Jae-in". Our meeting in Pyongyang feels like yesterday but about 100 days have already passed and now we are at the close of an unforgettable 2018'. Among other things, the Winter Olympics in South Korea's Pyeongchang in February saw a visit from Kim Yo Jong, the sister of the North Korean leader.
Kim and Moon met three times in 2018 and have made a series of goodwill gestures as well as vowing to resume economic cooperation when possible.
"Chairman Kim Jong Un emphasized the bold step the two Korean leaders have taken to overcome the long-pending confrontation between North and South to meet three times in one year alone", read the letter, according to Kim Eui-kyeom, spokesman for the South's presidential office.
The two neighbours technically remain at war after the 1950-53 Korean War ended with a ceasefire instead of a peace treaty. (EPA/AAP) A North Korean guard post is demolished inside the Demilitarized Zone separating the Koreas earlier this year.
The stalled negotiations had an impact on inter-Korean ties, including Kim's unrealized plan to visit Seoul this year as agreed their summit in Pyongyang in September.
In a rare letter to Seoul, Mr Kim said he wished to pursue peace between the countries and "solve the issue of denuclearising the peninsula together", a spokesman for Mr Moon said.
The move raises hopes for a positive message in his annual New Year's Day policy speech to be delivered on North Korea's state television.
Kim used his New Year's speech a year ago to initiate diplomacy with Seoul and Washington, which led to his meetings with Moon and a historic June summit with Trump.
Kim also wrote that he is ready to work with Moon to resolve the issue of denuclearizing the Korean Peninsula.