"Why would Kim Jong-un insult me by calling me 'old, '" tweeted the president.
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.
But in its November 11 statement, the North Korean Foreign Ministry said Trump's warnings "can never frighten us or put a stop to our advance", but rather "pushes us to speed up the efforts to accomplish the great cause of completing the state nuclear force".
The secretive kingdom's state-controlled media service told the North Korean people today how their former countrymen south of the Demilitarised Zone are suffering "untold pains under the U.S. occupation and domination".
The U.S.'s top diplomat said he'd look for a "relative period of quiet and an indication from Kim Jong Un himself that they would like to have some type of a meeting".
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; odd things happen in life" - he doesn't expect it to happen.
"If that did happen, it would be a good thing, I can tell you, for North Korea".
Mr Trump on Sunday also risked angering China, which he has sought as an ally in curbing North Korea's nuclear ambitions, by offering to mediate in a dispute over Beijing's claims to sovereignty in the South China Sea. "I don't know if it will but it would be very, very nice", he added.
'Whether it is arms, whether, whether it is cyber-crime, whether it is drugs they are constantly raising money to finance their nuclear program, ' Mr Turnbull said.
South Korea under President Moon Jae-in has differed from the U.S.in approach to the North Korea question.
North Korea conducted its sixth and most powerful nuclear bomb test on September 3, prompting another round of United Nations sanctions.
In a fresh show of force, three United States aircraft carriers, accompanied by South Korean warships, launched a joint naval drill yesterday.
In top-level talks in the Chinese capital on Thursday, Trump demanded China sever all financial links with North Korea, and said Chinese President Xi Jinping could come up with a solution as a "strong man".