The North Korean leader?s new year speech was also screened in the South
President Kim was giving his new year message broadcast on state television.
He said if Pyongyang?s conditions were met, he would even be prepared to hold a summit meeting with South Korean President Park Geun-hye.
South Korea said the move was ?meaningful?, and talks should include ?practical and frank discussions on all issues of mutual concern?.
?Our government hopes for dialogue between the South and North Korean authorities in the near future without limits on format,? said Unification Minister Ryoo Kihl-jae, quoted by the South?s Yonhap news agency.
On Monday, Mr Ryoo had called for dialogue to resume on issues including reunions for families separated by the Korean War, adding that he hoped North Korea would respond positively.
He offered to meet in Seoul, Pyongyang or any other South or North Korean city agreed with North Korean officials.
In his address, Mr Kim said the ?tragic? division of Korea could no longer be tolerable and acceptable.
?Depending on the mood and circumstances, there is no reason not to hold a high-level summit,? he said.
But later in his speech, Mr Kim condemned joint US-South Korean defence drills for deepening tensions on the peninsula.
?In a tense mood of such war-preparatory exercises, trust-based dialogue can?t be possible, and North-South relations can?t move forward,? he said.
The BBC?s Kevin Kim in Seoul says it is unclear how likely a meeting between the two Korean presidents would be, given the tensions.
The North has previously seen the South?s unification plans as an attempt to take it over.
The last formal high-level talks were in February 2014, leading to rare reunions for Korean families separated for over 60 years since the end of 1950-53 Korean War.
More talks planned in October were dropped after North Korea accused the South of not doing enough to stop activists sending anti-Northern leaflets across the border on balloons.
The two Koreas have technically been at war since the 1950-53 Korean conflict ended in an armistice rather than a peace treaty.