Trump quickly fired back: "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!"
But when it comes to the nuclear button, it seems Trump has been indulging in some "fake news" of his own, as despite claims of a "bigger and more powerful" nuclear button than Kim, there is actually no such thing.
Read more: Trump to Kim Jong-un: 'My nuclear button's bigger'
Instead, the President has access to a nuclear "football" which is essentially a briefcase containing communication tools and a "Denny's menu" of nuclear options that can be delivered at all times.
It's carried by a rotating group of military officers and follows him on Air Force One, around the golf course in a buggy and to private dinners at Mar-a-Lago.
He also has access to a card known as "the biscuit" containing nuclear codes that identify him as the President to military officials in the Pentagon, allowing him to launch a nuclear strike anywhere in the world within four minutes.
So while the proverbial "red button" does not technically exist, the President does have unique ability to launch a strike without any checks on his authority.
As nuclear expert Garrett Graff told news.com.au in September, "there is no part of this process where there is a second voice that has to say 'yes there is a good reason to launch nuclear weapons'."
"Or 'yes, I've doubled checked and the President isn't crazy drunk right now.' The President at whatever state he is in at any moment only has to confirm that he is the President of the US. That is the only check or balance on the entire system. From there the entire system is geared towards launching nuclear weapons as fast as possible."
Read more: Kim Jong-un's New Year message
Trump's comments yesterday were quickly mocked online with many speculating he was confusing it with the other button mentioned in a New York Times report used to summon a Diet Coke. Others backed up his hard line approach.
The tweets follow reports Kim Jong-un has ordered scientists to build North Korea's most powerful ballistic missile yet to be launched on the 70th anniversary of the regime in September 2018.
An anonymous defector from the rogue state with knowledge of the missile program told Japanese newspaper Asahi Shimbun, the plans were laid out in December to make a rocket fitted with re-entry capability in an upgraded version of a long-range Unha-3 missile that put North Korea's first satellite into orbit.
This week, North Korea has re-established a direct communication channel with the South ahead of potentially participating in the Winter Olympics next month, a move that has been welcomed by South Korea.