Asked to comment on Kim's speech, U.S. President Donald Trump said: "We'll see, we'll see", as he walked into a New Year's Eve celebration at his Mar-a-Lago resort in Florida.
Kim also emphasized that "the nuclear button is always on my desk", which he said is "not blackmail but reality".
North Korea has come under heavy sanctions imposed by the UN Security Council as it has accelerated efforts to expand its nuclear weapons and ballistic missile program.
Two North Korean figure skaters qualified to compete in the games and even though their country missed an October 30 deadline to confirm that it would send them, the International Olympic Committee has said it wants North Korea to participate and could still allow them to do so if the Koreas manage to work it out.
Kim said in a broadcast New Year's address Sunday that the entire U.S.is now in range of North Korea's arsenal.
The Winter Olympics will be held from 9-25 February, followed by the Paralympics from 9-18 March.
Kim said: "The Winter Olympic games that will be held soon in the South will be a good opportunity to display the status of the Korean nation, and we sincerely wish that the event will be held with good results". Last month, President Moon Jae-in urged Kim to hold off on testing any more nuclear warheads or ballistic missiles in the lead-up to Pyeongchang and proposed that the next round of South Korean-American military exercises be delayed.
In his speech Monday, Kim said the Olympics could provide a reason for officials from the neighbours "to meet in the near future". How things turn out, he said, rests on whether China puts more pressure North Korea.
Kim said in his speech Monday that North Korea past year achieved the historic feat of "completing" its nuclear forces.
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. "He's developing nukes while trying to weaken global pressure and the South Korea-U.S. military alliance and get worldwide sanctions lifted", said Shin Beomchul of the Seoul-based Korea National Diplomatic Academy.
Daniel Russel, the top USA diplomat for East Asia until last April and now at the Asia Society Policy Institute, said there was an argument to be made to encourage North Korea's Olympic participation but that it should not be taken too far.
New Year's Eve wind chill in Detroit forecast below zero
Thursday and through the weekend, high pressure in the upper-levels will take over and a warming trend will take over. Boston hovers around 5 degrees at midnight and not that far off the record low of -3 the morning of New Years Day .