Trump has called Kim "rocket man" and Kim has called Trump a "mentally deranged US dotard", a "gangster fond of playing with fire", and "a frightened dog".
Trump goes on to say sarcastically, "Oh well, I try so hard to be his friend - and maybe someday that will happen!"
Aggressive rhetoric from North Korea is nothing new, and on Saturday the secretive nation described Trump as a "destroyer" who "begged for nuclear war" as the President continued his tour of Asia.
Trump told an audience of CEOs at the Asia-Pacific Economic Cooperation summit in Danang, Vietnam, that the region's future "must not be held hostage to a dictator's twisted fantasies of violent conquest and nuclear blackmail". The statement was issued in response to a speech delivered by Trump in South Korea's National Assembly on Tuesday, where he stressed that the U.S. should not be underestimated, and that North Korea's new weapons might be putting the regime in more, rather than less danger.
Trump's visit to Beijing last week Xi reiterated that China would strive for the denuclearization of the Korean peninsula but offered no hint it would change tack on North Korea, with which it fought side by side in the 1950-53 Korean war against US -led forces. "And do not try us". He said "every single step the North Korea regime takes toward more weapons is a step it takes into greater and greater danger".
Trump and Kim have engaged in name-calling in recent months as tensions between the two countries increase and North Korea continues expanding its nuclear program.
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Asked if he could see himself being friends with Kim, Trump said: "That might be a odd thing to happen but it's a possibility".
Nuclear-armed North Korea regularly denounces such military drills as rehearsals for invasion and sometimes conducts its own military manoeuvres or missile tests in response.
He said he doesn't know if that friendship will develop, but added it would be "very, very nice if it did".
This isn't the first time Trump and Kim have traded barbs.
Tensions have been running high since North Korea test-fired two intercontinental ballistic missiles in July, conducted its sixth and most powerful nuclear test on September 3 and flew ballistic missiles over Japan on August 29 and September 15.