In an effort to rally global pressure against North Korea's nuclear weapons program, Trump had said during his speech in South Korea on Tuesday, "Do not underestimate us".
Though an armistice ceased fighting in that conflict, no peace treaty has ever been signed, and North Korea still tensely face off with South Korean and United States forces at the demilitarized zone that splits the Korean peninsula at the 38th parallel.
North Korea, a country in a "time warp" that feeds paranoia at home with propaganda about a hostile world, could secure its future by laying the groundwork for talks on its nuclear weapons program, according to South Korea's foreign minister.
With tensions running high over North Korea, Trump mocked his nemesis Kim, deviously calling him "short and fat", after Pyongyang's spokesman dubbed Trump a "dotard".
Mr Trump's own dispute with North Korea continues to escalate, at least in the insults traded across Twitter and in official statements.
At a news conference a few hours later, Trump said that while it would be "a good thing" if he and Kim became friends - "I think anything is a possibility; unusual things happen in life" - he doesn't expect it to happen.
President Trump responded to the latest round of name-calling from North Korean leader Kim Jong Un on Saturday, saying "I try so hard to be his friend" and would "never call him "short and fat". "If it did happen, it would be a good thing for North Korea, and a good thing for lots of other places and good for the world". "I don't know that it will, but it would be very, very nice if it did".
Speaking to reporters aboard Air Force One, Mr Trump said Mr Putin had again vehemently insisted - this time on the sidelines of an economic summit in Vietnam - that Moscow had not interfered in the 2016 U.S. elections. This came after the Asian dictator called Trump "old".
Trump, who is 71 years old, made the comment after attending the Asia-Pacific Economic Cooperation (APEC) summit in Vietnam. "The president wants to build upon that to really build our relations with Asean".
Also speaking on Sunday at a meeting in Hanoi with Vietnam's president, Tran Dai Quang, Trump offered to mediate in a dispute between Vietnam and China over control of the South China Sea.
"As to whether I believe it or not, I'm with our agencies, especially as now constituted", Trump said.