Trump launched the highly personal missive on Twitter hours after his ambassador to the UN Nikki Haley described proposed dialogue between the two Koreas as a "band-aid" and said Washington would never accept a nuclear-armed Pyongyang.
Trump said: "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!"
The tweet was in reference to Kim's annual New Year address in which he warned he has a, but sweetened his remarks by expressing an interest in dialogue and taking part in the Pyeongchang Games next month.
South Korea has responded positively to Kim's overture, suggesting January 9 as a date for rare talks aimed at easing tensions on the Korean Peninsula.
But the US questioned whether talks could be taken seriously.
North Korea has rattled the international community in recent months with multiple missile launches and its sixth and most powerful nuclear test -- purportedly of a hydrogen bomb.
It has shrugged off a raft of new sanctions and heightened rhetoric from Washington as it drives forward with its weapons program, which it says is for defense against US aggression.
Pyongyang claims it needs nuclear weapons to protect itself from a hostile Washington and has striven to create a warhead capable of targeting the US mainland with an atomic warhead.
Personal insults
Earlier, South Korea's President Moon Jae-In welcomed Kim's comments as a "positive response" to Seoul's hopes that the Pyeongchang Olympics would be a "groundbreaking opportunity for peace"
The South's unification minister Cho Myoung-Gyon told a press conference Tuesday that Seoul was "reiterating our willingness to hold talks with the North at any time and place in any form."
"We hope that the South and North can sit face to face and discuss the participation of the North Korean delegation at the Pyeongchang Games as well as other issues of mutual interest for the improvement of inter-Korean ties," he added.
The Koreas, divided by a Demilitarized Zone since the end of the 1950-53 Korean War, last held high-level talks in 2015 to try to ease tensions.
But Haley, the US Ambassador to the UN, told reporters Washington could not take the talks seriously "if they don't do something to ban all nuclear weapons in North Korea."
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She said: "We consider this to be a very reckless regime. We don't think we need a band-aid and we don't think we need to smile and take a picture.
"We think we need to have them stop nuclear weapons and they need to stop it now," she said, warning: "We will never accept a nuclear North Korea."
Trump's tweet was also the latest in a series of personal insults the two leaders have traded since the US president took office a year ago.
Trump has mocked Kim as "fat" and a "little rocket man." Kim, for his part has described Trump as a "mentally deranged US dotard."
