The North Korean leader sent Moon a personal letter of well wishes on Sunday, expressing a willingness to meet often in 2019 to advance peace talks and achieve "denuclearization on the Korean peninsula", Moon spokesman Kim Eui-keum said.
"There will still be a lot of difficulties ahead", Moon said in his message.
The stalled negotiations have had an impact on inter-Korean ties, with Kim failing to visit Seoul this year as agreed to at an inter-Korean summit in Pyongyang in September.
One thing North Korea is being a bit more direct on is wanting more summits between Kim and South Korea's President Moon Jae-in. "However, our hearts will become more open if we put in that much effort".
The US president praised a "warm letter" from Pyongyang in September, although no date has been set for another meeting.
The rivals have also taken steps to reduce their conventional military threat, such as removing mines and firearms from the border village of Panmunjom, destroying some front-line guard posts and creating buffer zones along their land and sea boundaries and a no-fly zone above the border.
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'Chairman Kim said he will keep a close eye on the situation and expressed strong will to visit Seoul.
Moon thanked him for the letter, tweeting that the North Korean leader "again made clear" that he would act on his agreement with the US and South Korea.
Despite these setbacks, another summit between Mr Trump and Mr Kim could happen in 2019.
The letter comes days before Kim is expected to address North Koreans in a New Year's speech that North Korean leaders traditionally use to announce major policy decisions and goals.
"The New Year's Day speech will likely address implementation of peace and prosperity, the denuclearization issue and maintaining the inter-Korean relations", South Korean Unification Ministry spokesman Baik Tae Hyun told a briefing.
Earlier this month, North Korea told the US that sanctions and pressure won't work to force Pyongyang into action on its nuclear program.
Talks between the US 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.
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