Trump has traded insults and threats with Kim in the past amid escalating tensions over Pyongyang's nuclear and missile programs as North Korea races toward its goal of developing a nuclear-tipped missile capable of reaching the United States - something Trump has vowed to prevent.
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.
In a response to North Korea calling Trump's speech in South Korea "reckless remarks by an old lunatic", Trump tweeted from Hanoi on Sunday morning: "Why would Kim Jong-un insult me by calling me 'old, ' when I would NEVER call him 'short and fat?'"
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.
One of Trump's tweets read: "Why would Kim Jong-un insult me by calling me 'old, ' when I would NEVER call him 'short and fat?' Oh well, I try so hard to be his friend - and maybe someday that will happen!"
"I really believe that it makes sense for North Korea to come to the table and to make a deal", Trump said in a press conference with South Korean President Moon Jae-in.More news: Turkey first to deliver relief supplies to quake-hit northern Iraq
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Asked if he could see himself being friends with Kim, Trump said: "That might be a unusual thing to happen but it's a possibility".
It comes on the same day Rodong Sinmun, the regime-run newspaper, warned the North would "surely win in the showdown" with the US and South Korea. "If that did happen, it would be a good thing for, I can tell you, for North Korea".
The North Korean foreign ministry reportedly called Trump's Asia trip a "a warmonger's trip for confrontation with our country".
While North Korea has launched around two dozen missiles this year, including an intercontinental ballistic missile that can strike parts, if not most, of the continental US, and tested a staged thermonuclear weapon with an explosive yield far greater than anything the regime has previously tested, the North has not engaged in major provocations in almost two months.
Speaking later in Vietnam's capital, Hanoi, Trump said "it would be very, very nice" if he and Kim became friends.
Also on the trip, Mr Trump met with Russian President Vladimir Putin several times on the margins of a summit in Vietnam.