Mr Trump's tweet is the latest contribution to the bickering, increasingly personalised feud between the nuclear-armed leaders.
Mr Kim threatened earlier this week that his nuclear launch button was "always on my table".
Unsurprisingly, Mr Trump's unorthodox words sent social media into a frenzy.
It ended a quick-fire day of tweeting that included taking credit for a lack of airplane crashes, announcing awards for "corrupt media", and threatening to pull aid from Palestinians who do not show "appreciation or respect".
Mr Trump's latest comment states the obvious: any US president has immediate access to the nuclear codes and the US has the world's biggest nuclear arsenal.
North Korea claims it has nuclear weapons and could attack the US, but while analysts agree that Pyongyang has nuclear weapons, it is not clear whether it has the technology to use them in anger.
North Korean Leader Kim Jong Un just stated that the “Nuclear Button is on his desk at all times.” Will someone from his depleted and food starved regime please inform him that I too have a Nuclear Button, but it is a much bigger & more powerful one than his, and my Button works!
— Donald J. Trump (@realDonaldTrump) January 3, 2018
(BBC News)
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