Talks between the USA and North Korea have made little progress since Trump and Kim signed an agreement in June to "work toward complete denuclearization of the Korean Peninsula", without defining the term or setting a deadline.
It is Kim's first letter to Moon in 10 months.
The report did not include details about the message but said it related to US-North Korea talks, and was conciliatory in tone.
North Korea's leader, Kim Jong-un, has vowed in a rare letter to meet the South's president, Moon Jae-in, "frequently" next year to discuss denuclearisation of the Korean peninsula, Moon's office said Sunday.
"There is a discourse channel between North Korea and the United States through which they trade dynamic correspondence, yet I can't know whether it appeared as letter or something different", Moon's representative told a news preparation on Monday.
The spokesman said the office summarized Kim's two-page letter and paraphrased it in different words and what he said wasn't a direct quote from the North's leader as it is a violation of diplomatic custom.
"Even though there will be a lot of difficulties going forward, our hearts will be opened to each other depending on how much effort we make", he said in a statement posted on Facebook.
The South Korean President said he hoped to see Kim in the new year, and that his welcome to the country "remains unchanged".
Kim said he was willing to meet the USA president at any time, mixing in conciliatory messages with his warnings of conflict to Washington and Seoul in his annual speech where he laid out policy priorities for his isolated and impoverished country.
Kim also lamented not being able to visit Seoul before the end of the year and reaffirmed his intention to do so while "closely watching the situation", the spokesman said.
The leader of the isolated North met Moon three times this year, twice at the border truce village of Panmunjom and once in Pyongyang, as a reconciliatory push gathered pace.
Hours after its disclosure on Sunday, Moon hailed Kim's renewed commitment to denuclearization. With global sanctions squeezing the country's trade flow and cash supply, any clues about the North Korean economy or Kim's growth strategies will be closely examined.
Denuclearization talks, however, have been in a stalemate as Pyongyang wants sanctions relief for subsequent steps following the summit, while Washington is asking for more concrete measures before making major concessions.