The US President-elect, who will be inaugurated on 20 January, put out a tweet in response to a claim by Kim Jong-Un that his country is close to testing an intercontinental ballistic missile (ICBM).
ICBMs can travel up to 10,000km (6,200 miles). The United States is around 9,000km (5,500 miles) from North Korea.
The tweet prompted South Korea to say it shows Mr Trump is aware of the threat posed by the North’s nuclear programme and will not waver from a policy of sanctions against the isolated country.
Mr Trump tweeted: “North Korea just stated that it is in the final stages of developing a nuclear weapon capable of reaching parts of the U.S. It won’t happen!”
South Korea’s foreign ministry said Mr Trump’s comment – his first mention of the North Korean nuclear issue since the US election in November – could be interpreted as a “clear warning”.
Foreign ministry spokesman Cho June-hyuck said: “Because of our active outreach, President-elect Trump and US officials are clearly aware of the gravity and urgency of the North Korean nuclear threat.
“They are maintaining an unwavering stance on the need for sanctions on North Korea and for close cooperation between South Korea and the US.”
Mr Trump is yet to outline a policy on North Korea, but during the election campaign he suggested he would be willing to talk to Kim if he had the opportunity.
The US’ policy for several years has been that North Korea must disarm first before talks can take place.
When North Korea carried out two nuclear tests and various missile launches last year, the US and South Korea responded with even tougher sanctions.
In a subsequent tweet, Mr Trump went on to criticise China for not doing enough to stop the North’s nuclear programme.
He tweeted: “China has been taking out massive amounts of money & wealth from the U.S. in totally one-sided trade, but won’t help with North Korea. Nice!”
China responded by saying that its hard work in trying to ensure the denuclearisation of the Korean peninsula is obvious to all.
Foreign ministry spokesman Geng Shuang also urged the US to appreciate the sensitivity of the Taiwan issue after Mr Trump left open the possibility of meeting Taiwan’s president.
Such a move would be a reversal of Washington’s longstanding “One China” policy, which means the US recognises Beijing’s assertion that Taiwan is a part of its territory.