It comes as tension in the region has ratcheted up after Pyongyang conducted a series of ballistic missile tests and a pair of nuclear tests over the last year.
“We hope the two sides sit down for frank talks,” the South’s unification minister, Cho Myoung-gyon, told a news conference.
He also proposed a televised meeting of the two governments in Panmunjom – a village on the border between the two nations.
His comments came after a closely watched new year address in which Mr Kim said he was willing to send a delegation to the Olympics.
However, he also repeated nuclear threats against the United States.
Mr Kim’s overture was welcomed by the South’s government, which is led by liberal President Moon Jae-in, who favours dialogue with his northern neighbour to ease its nuclear threat.
He also hopes to use the Winter Olympics to improve inter-Korean ties.
He said Mr Kim had responded positively to South Korea’s proposed dialogue between the two nations and had ordered officials to look into restoring talks between the two nations.
He also told them to look into whether North Korean athletes could compete in February’s winter games.
Some commentators suggested Mr Kim may be trying to drive a wedge between Seoul and Washington as a way to ease international sanctions.
If there are talks, they would be the first formal dialogue between the Koreas since December 2015.
Relations between the pair have deteriorated over the last year as North Korea pushed forward with its nuclear weapons programme.
In September, the secretive Communist state carried out its sixth and most powerful nuclear test.
It also conducted three intercontinental ballistic missile tests, some of which were launched over the Japanese mainland, as part of its drive to develop a nuclear missile capable of reaching anywhere in the United States.
The United Nations subsequently imposed tougher sanctions on the pariah state.
Were the talks to happen, it could also weaken US-led international pressure on the North.
North Korea has made it clear that it has no intention of accepting international calls for nuclear disarmament and instead wants to bolster its weapons arsenal in the face of what it calls increasing US threats.
In his new year address, Mr Kim said the United States should be aware that his country’s nuclear forces are now a reality, not a threat.
He said he has a “nuclear button” on his office desk and warned that “the whole territory of the US is within the range of our nuclear strike”.
But he struck an unusually conciliatory note with the the South, saying he wanted a “peaceful resolution with our southern border”.
He called for improved ties and a relaxation of military tensions with South Korea, saying the Winter Olympics could showcase the status of the Korean nation.
He nonetheless maintained his call for South Korea to stop its annual military exercises with the United States, which he calls a rehearsal for war with his country.
The new year address is an annual event in North Korea and is watched closely for indications of the direction and priorities Mr Kim may adopt in the year ahead.
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