"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!" Trump tweeted.
The threatening tweet followed a New Year's Day address from Kim, who said that North Korea's nuclear weapons can reach anywhere in the United States and threatened that he has a nuclear button on his desk.
Much of the world has been on edge as the two leaders have threatened one another's countries with nuclear weapons repeatedly over the last year. The United States and North Korea fought a war in the 1950s.
One strategist told CNBC on Tuesday that Trump actions against North Korea could result in a cold war between the U.S. and China.
David Roche, president and global strategist at Independent Strategy, said such a turn of events between two economic and military powerhouses would have global implications for worldwide trade and investment.
Earlier Tuesday, Trump said that sanctions on North Korea were "beginning to have a big impact," following word of proposed talks between Pyongyang and Seoul about next month's Winter Olympics in South Korea.
Experts point out that the so-called "Nuclear Button" that Trump mentioned doesn't literally exist, though any U.S. nuclear actions would originate from the White House.
— CNBC's Jacob Pramuk and Yen Nee Lee contributed to this report.