The PyeongChang Winter Olympic and Paralympic Games were scheduled to be held in February in South Korea's namesake county of PyeongChang in the eastern Gangwon province.
But many North Korea watchers say the young leader's two-track approach is a ploy to shake the South Korea-U.S. alliance.
He added that the North did not show any reaction to a call by a South Korean liaison official at the truce village of Panmunjom earlier in the day.
U.S. President Donald Trump and first lady Melania Trump, with their son Barron, arrive for a New Year's Eve party at his Mar-a-Lago club in Palm Beach, Florida, U.S. December 31, 2017.
"An improvement in inter-Korean relations may not go a separate way from (efforts) to resolve the North Korean nuclear issue, so I ask the foreign ministry to closely consult with our allies and the worldwide community so that we may push for an improvement in inter-Korean relationship and the resolution of the North Korean nuclear issue at the same time", he said.
Choi Moon-soon, governor of Gangwon Province where the Pyeongchang Olympics is set to be held, has proposed South Korea would send cruise ships to bring North Korean athletes and officials to Pyeongchang, according to South Korean media.
Referring to the Olympics, he said later: "We're willing to take necessary steps including sending our delegation, and for this, the authorities from the North and South could urgently meet".
As well as Kim's overtures to his Southern neighbors, he also announced plans to start mass-producing nuclear warheads as he delivered a televised New Year's Day speech.
South Korea said it welcomed Kim's offer.
Kim said that rather than encouraging USA measures that "threaten the security and peace of the Korean peninsula", Seoul should instead respond to overtures from the North, and "stop nuclear war exercises with foreign forces". He said the Pyeongchang Winter Olympics would be a good opportunity to showcase the status of the Korean nation.
The tensions around North Korea's missile and nuclear weapons program persist due to the repeated nuclear tests and missile launches carried out by Pyongyang in violation of the UN Security Council (UNSC) resolutions.
Kim's customary New Year's speech is closely watched for indications of the policy direction the unpredictable and reclusive leader is likely to pursue in the coming year.