The Korean Central News Agency referred to Trump on Saturday as a "destroyer" and a "dotard" who "begged for nuclear war".
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.
The president's remarks in a wide-ranging talk with reporters on Air Force One en route to Hanoi on Saturday and in his Sunday morning tweets have sharply punctured the careful messaging he and his aides had sought to deliver on a five-nation, 12-day trip through Asia.
Trump was responding to North Korea calling his South Korea speech days earlier: "reckless remarks by an old lunatic".
On Saturday, Kim's government responded by accusing Trump of trying to demonize North Korea, keep it apart from the worldwide community and undermine its government.
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".
North Korea has dominated Mr Trump's Asia trip so far to Japan, South Korea, China and Vietnam.
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.
"If it did happen it could be a good thing I can tell you for North Korea, but it could also be good for a lot of other places and be good for the rest the world".
The Trump administration has said all military options remain on the table when dealing with the North Korean threat, but top US officials have consistently emphasized the U.S.is pursuing a diplomatically led effort, including additional economic pressure.
"Every step you take down this dark path increases the peril you face".
For the last months, the two leaders have been on two ends of a bitter war of words prompted in part by Pyongyang refusal to cease ballistic missile tests and the development of nuclear capabilities.
He said he doesn't know if that friendship will develop, but added it would be "very, very nice if it did".
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.