In his own message, Moon thanked Kim for his "warm" letter and said he was still invited to South Korea, according to the AP.
In an attempt to start working-level negotiations, the USA said it would review its policy on humanitarian aid to North Korea.
It was sent days before Kim delivers his annual New Year's Day speech, which will be examined for signs of whether the USA and North Korea will continue with their rapprochement or drift back toward confrontation.
"[Chairman Kim] redefined his standpoint in actively carrying out agreements in the inter-Korean summit as well as the US-DPRK summit", Moon wrote, continuing, "I welcome chairman Kim's intentions to meet several times in the new year to find solution for denuclearization and to solve practical problems in achieving peace and prosperity".
Cheong Wa Dae added that Moon will likely send a response to Kim in a letter sooner.
Moon added that he witnessed a sense of frustration over Kim not having been able to visit Seoul, which the two leaders agreed to during the third inter-Korean summit in Pyongyang this September.
The North Korean leader called on South Korea to halt joint military exercises with the US, placing new pressure on the alliance. Officials in neighboring countries follow the speeches closely to obtain overall information about the country and how it will act on diplomatic issues such as its nuclear weapons program.
Kim's letter has renewed hopes for inter-Korean relations in 2019 - ending weeks of lackluster communication between the two Koreas as the North's denuclearization talks with the USA have gone off the boil.
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Kim highlighted recent exchanges with socialist countries, speaking on his trips to China and the Cuban president's visit to North Korea.
Meanwhile Ted Poe, the chairman of the U.S. House Subcommittee on Terrorism, Nonproliferation, and Trade, told Congress that the West's maximum pressure policy "appears at risk due to South Korea's dovish president".
The U.S. has been critical of North Korea recently, raising internationally the issue of human rights abuses.
"There is a dialogue channel between North Korea and the United States through which they exchange active communication, but I can not know whether it took the form of letter or something else", Moon's spokesman told a news briefing yesterday.
Kim had yet to confirm the timing of his unprecedented visit to Seoul, the South Korean capital, but the groups are already preparing for his arrival.
The conciliatory mood has continued to grow ever since Kim extended a rare olive branch to South Korea in his New Year speech at the start of this year to express his willingness to improve inter-Korean relations.
"President Moon must not be deceived by Kim's small gestures of goodwill and prematurely fold", Poe warned. The North sent a delegation, including Kim's sister Kim Yo-jong, to the PyeongChang Winter Olympics held in February.