The North has yet to make a statement on the meeting, announced by the South.
Earlier, the White House said the meeting would not take place unless Pyongyang took "concrete actions".
US media report that Mr Trump made the decision to meet without consulting key figures in his administration, who are now scrambling to catch up.
No sitting US president has ever met a North Korean leader.
Mr Trump stunned observers when he agreed to the summit following an invitation delivered by South Korean envoys.
Confusion mounted when Mr Trump's own press secretary, Sarah Sanders, told reporters that North Korea has "promised to denuclearise". She added: "We're not going to have this meeting take place until we see concrete actions."
The top US diplomat, Secretary of State Rex Tillerson, was on his first official trip to Africa when the announcement was made.
He told reporters on Friday the decision to meet Mr Kim was one "the president took himself".
"I spoke to him very early this morning about that decision and we had a very good conversation," Mr Tillerson added.
On Saturday, Mr Tillerson cancelled all his official events in Kenya with aides saying the secretary was "not feeling well after a long couple days working on major issues back home such as North Korea".
South Korean envoys - who recently met with Mr Kim in Pyongyang - have said North Korea is "committed to denuclearisation" as an end goal, but they have not said this would start before a meeting with the US.
Instead, North Korea is understood to have agreed to halt its testing programme as negotiations continue.
US Vice-President Mike Pence has pledged to maintain pressure on North Korea, and Mr Trump spoke with Chinese president Xi Jinping on Friday to agree to maintain sanctions for the time being.
Chinese state media said the meeting resulted from Chinese efforts, with the Communist Party's newspaper the People's Daily saying the US "profusely thanked and put high importance on China's important role".
There has been no mention of any meeting in North Korean media.
An initial statement from the South Korean delegation said the meeting would take place by May - but no place or date has officially been set.
The Korean border's demilitarised zone (DMZ) and Beijing are seen as possible venues.
How did we reach this point?
Kim Jong-un unexpectedly used his New Year's message to reciprocate a offer of talks made by the South last year. This led to North Korea sending a delegation to the Winter Olympics in the South.
After the Games, Then, South Korean envoys met Mr Kim in Pyongyang this week. The envoys then travelled to Washington to brief Mr Trump.
Speaking outside the White House after the meeting, South Korean National Security Adviser Chung Eui-yong said Mr Kim was prepared to sit down with the US president and was now "committed to denuclearisation".
In a statement sent to the Washington Post, North Korea's UN ambassador said the "courageous decision" of Mr Kim would help secure "peace and stability in the Korean peninsula and the East Asia region".
However, the North has halted missile and nuclear tests during previous talks, only to resume them when it lost patience or felt it was not getting what it demanded, analysts say.
Some expressed concern the Trump regime could "fall into the North Korean trap" of granting concessions with nothing tangible in return.