"The United States should know that the button for nuclear weapons is on my table", he said during the nationally televised speech, according to a provisional translation by The Associated Press.
After welcoming Kim's address, South Korean President Moon Jae-in had asked his government earlier in the day to move as quickly as possible to bring North Korea to the Olympics.
This shows that Kim does not intend to abandon the development of nuclear weapons and intercontinental ballistic missiles (ICBMs).
Lee Hee-Beom, the head of the Pyeongchang Organising Committee for the Olympic Games (POCOG) told AFP: "We actively welcome the North's suggestion that it is willing to engage in talks for the participation in the Olympics".
This week's exchanges follow a year dominated by fiery threats form Kim and U.S. President Donald Trump who vowed to destroy North Korea if threatened, even as U.S. diplomats pushed for a diplomatic solution.
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"The upcoming Olympics provide an important opportunity to break the ice and to begin discussions with the North Koreans on mutual steps that reduce the chances of miscalculation and war", he added.
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", Kim said. The North sees the regular war drills between the South and the United States as preparations for war.
Cho told a media conference he expects the two Koreas to focus on bringing a North Korean delegation to the Pyeongchang Winter Olympics, which begin next month, should the talks be held on January 9.
The US has no reason to oppose improved inter-Korean relations being used to open the door to North Korea-US negotiations aimed at resolving the nuclear issue.
North Korea tested intercontinental ballistic missiles and conducted its sixth and most powerful nuclear test in September in defiance of global warnings and sanctions, raising fears of a new conflict on the Korean peninsula.
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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.
The venues for the Games lie just 80 kilometres (50 miles) south of the tense border with the North and the build-up to the event has been overshadowed by tensions running high over the North's escalating nuclear and missile tests.
But Kim, dressed in a Western-style gray suit and a tie, also offered a potential olive branch to Seoul saying it is imperative to lower military tensions on the Korean Peninsula and improve ties with the South.
Despite increased global sanctions imposed over the weapons programme, North Korea made progress in areas like fabrics, shoes and tractors, Kim said.
"We welcome it", a statement said.
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Harry Kazianis, director of defense studies at Washington's conservative Center for the National Interest, said that if North Korea did participate in the Olympics, there could be a lull in tensions, but only a brief one.