In an extraordinary turn of events, he confirmed on Twitter that a meeting was being planned despite the two leaders trading insults for months and economic sanctions on North Korea being ramped up in response to its nuclear weapons programme.
Mr Trump had previously threatened “fire and fury” in response to threats from North Korea. Mr Trump also referred to the North Korean leader as “Little Rocket Man”.
But today, the US leader tweeted: “Kim Jong Un talked about denuclearisation with the South Korean representatives, not just a freeze.
“Also, no missile testing by North Korea during this period of time. Great progress being made but sanctions will remain until an agreement is reached. Meeting being planned!”
It comes after South Korea’s national security adviser Chung Eui-yong announced at the White House Mr Kim’s desire to meet with Mr Trump following talks in Pyongyang on Monday.
“He (Mr Kim) expressed his eagerness to meet President Trump as soon as possible,” said the South Korean official. “President Trump appreciated the briefing and said he would meet Kim Jong Un by May to achieve permanent denuclearisation.”
Mr Chung also said Mr Kim had vowed that his country would “refrain from any further nuclear or missile test” and that he was “committed to denuclearisation”.
He said North Korea understood routine military exercises carried out by the US and South Korea must continue and that there would be continuing pressure on the country until it makes “concrete actions”.
The White House confirmed a meeting between the two would be held “at a place and time to be determined”.
South Korean President Moon Jae-in said a meeting between Mr Trump and Mr Kim would be a “historical milestone” that will put the denuclearisation of the Korean Peninsula “really on track”.
Mr Moon also complimented Mr Trump for accepting Mr Kim’s invitation for a summit, saying Mr Trump’s leadership will be praised “not only by the residents of South and North Korea but every peace-loving person around the world”.
No serving US President has ever met with a North Korean leader and the two countries do not have formal diplomatic relations.
The nations are still technically at war following the 1950-1953 Korean War, which ended with an armistice and not a peace treaty.
The latest attempt to open a dialogue between the two countries comes after Mr Kim hit out at the US in his New Year’s address and referred to the “nuclear button” on his office desk.
Mr Trump later tweeted that he had a nuclear button too that was a “much bigger and more powerful one than his, and my button works!”
Former US ambassador to South Korea Christopher Hill said North Korea’s invitation was “quite extraordinary”.
He told Sky News: “I spent four years negotiating with the North Koreans as part of the six-party talks and I can tell you we never had a moment anything close to this.
“Certainly, the North Koreans have invited American heads of state before but this is the first time an American head of state has said ‘sure I’ll go’ – so this is really quite extraordinary.”
Sky News correspondent Mark Austin said the timing of the meeting was “bizarre to say the least”.
“He’s had this invitation today, he’s accepted it today and he’s said he’s going to meet by May. Now, by any standards, that is an incredibly quick decision to make on something as important as this,” he said.
“Most summit meetings take place after months and months of preparatory meetings by lower level officials – now there’s not time for that in this case.”
He added: “It may be that Donald Trump is anxious to get this photo call and get things moving… It will take months and months and months to come to any sort of deal with North Korea on this. The Americans will go in with acute suspicion and they’ll be wondering why on earth there’s been this sudden turn around by Kim Jong Un.
“It’s all very odd and we should treat it with suspicion.”
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Leading Republican senator Lindsey Graham hailed Mr Trump’s North Korea strategy and warned Mr Kim not to “try and play” the US President when they meet, with it being “the worst possible thing you can do”.
“If you do that, it will be the end of you – and your regime,” he added in a statement on Twitter.