On Sunday, the office of South Korea's President Moon Jae-in said Kim had sent a letter to his counterpart in Seoul, saying he wanted to hold more inter-Korean summits next year.
The stalled negotiations had an impact on inter-Korean ties, including Kim's unrealised plan to visit Seoul this year as agreed at their summit in Pyongyang in September.
He expressed "strong resolve" to make it happen while monitoring the situation.
"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.
He promised to make the trip after Mr Moon visited Pyongyang in September. Instead of standing on a podium, Kim delivered a speech sitting on a couch in a Western-style room, illustrating an image of an approachable leader in a fireside chat. It was the first time a member of the ruling Kim dynasty had visited the South since the 1950-1953 Korean War.
Cheong Wa Dae also said Kim assessed that the two leaders made bold decisions through three meetings and they helped ease the longtime tension between the two Koreas.
But the USA maintains it will keep sanctions in place against North Korea. The aide declined to reveal further details about how the letter was delivered to the presidential office for "a security reason".
An unexpected letter from the North.
Uploading the two-page letter on his Facebook and Twitter accounts, .
It was nearly exactly one year ago, in the wake of Kim's 2018 policy speech, that Trump threatened the North Korean dictator and said his nuclear button was "much bigger and more powerful". The U.S. granted an exemption on sanctions against the North to give it access to materials needed for the projects.
Neither the U.S. State Department nor the U.S. Embassy in Seoul had an immediate comment about the report of Kim's message to Trump when contacted by media.
Following the shifts, critics expect the North could send more flexible messages about resuming talks with neighboring countries, and especially with the U.S.
The summits are to achieve denuclearisation of the peninsula.
Moon - a dovish former human rights lawyer - has advocated dialogue to push the impoverished but nuclear-armed North toward denuclearisation.
Kim and Trump are trying to arrange a second summit in early 2019.
For Kim, working with South Korea has been a lot more productive than working with the US.