In a statement lashing out at Trump on Saturday, North Korea also referred to him as a "dotard," a word meaning a very old person, and one the reclusive nation has used on him in the past.
Trump fired back hours later.
"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!" Trump tweeted.
Asked about the possibility of becoming friends with the North Korean leader during a Sunday joint news conference in Vietnam, Trump did not rule it out.
"Strange things happen in life. That might be a strange thing that happens. But it is certainly a possibility," Trump said. "If that did happen, it would be a good thing for, I can tell you, for North Korea. But it would also be good for lots of other places, and it would be good for the world."
Trump is in Vietnam as part of a nearly two-week tour of Asia
, and North Korea's nuclear weapons program has been a major talking point with his Japanese, South Korean and Chinese counterparts.
"The weapons you are acquiring are not making you safer, they are putting your regime in grave danger," Trump said about North Korea during an address at South Korea's National Assembly in Seoul. "Every step you take down this dark path increases the peril you face."
He called on nations to stop financing and trading with North Korea.
"Together we have in our power to finally liberate this region and the world from this very serious nuclear menace, but it will require collective action, collective strength and collective devotion to winning the peace," Trump said.
North Korean officials described Trump's trip as "nothing but a business trip by a warmonger to enrich the monopolies of the US defense industry."
"Trump, during his visit, laid bare his true nature as destroyer of the world peace and stability and begged for a nuclear war on the Korean peninsula," the nation's foreign ministry said in a statement.
The harsh words between the leaders have escalated in recent months after North Korea threatened to launch missiles toward the US territory of Guam
Trump attempted a symbolic stare-down of Kim this week at the heavily fortified border that separates North and South Korea, but heavy fog forced the cancellation of his plans
The US on Saturday began a three-carrier strike force exercise in the Western Pacific. It involves the USS Ronald Reagan, the USS Nimitz and the USS Theodore Roosevelt. The exercise, set to continue through Tuesday, will involve operations showing the Navy's ability to operate multiple carrier strike groups as a coordinated effort, it said in a release.
The strike force plans to conduct air defense drills, sea surveillance, defensive air combat training as well as other maneuvers.
This is the first time three carrier strike groups will operate together since 2007 during an exercise off the coast of Guam, according to the Navy.
Trump continued to push his harsh rhetoric for North Korea on Sunday, telling reporters that "the world has been provoked" by the rogue nation.
"We want progress not provocation," Trump said. "We have been provoked. The world has been provoked."
He added that the world needs peace, not war.