"North Korea just stated that it is in the final stages of developing a nuclear weapon capable of reaching parts of the US," Trump tweeted. "It won't happen!"
Trump's tweet came a day after North Korean leader Kim Jong-Un appeared to try to pressure the incoming president by announcing his country is in the "final stages" of developing an intercontinental ballistic missile (ICBM).
Kim also said his country had significantly bolstered its nuclear arsenal last year.
"President-elect Trump's message is significant since it is his first mention of North Korea's nuclear programme and can be seen as a clear warning," South Korean foreign ministry spokesman Cho June-Hyuck told a briefing.
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Cho said the incoming US administration was clearly aware of the "gravity and urgency" of Pyongyang's nuclear threat thanks to South Korea's "active outreach".
US policy on the North would remain largely unchanged, he said.
"They are maintaining an unwavering stance on the need for sanctions on North Korea," Cho said.
Analysts are divided over how close Pyongyang is to realising its full nuclear ambitions, especially as it has never successfully test-fired an ICBM.
However, it carried out two nuclear tests and numerous missile launches last year alone in pursuit of its oft-stated goal -- developing a weapons system capable of hitting the US mainland with a nuclear warhead.