The shock announcement was made by senior South Korean officials in Washington, who passed on a letter from the North Korean leader.
They said Mr Kim had also agreed to halt nuclear and missile tests and was "committed to denuclearisation".
It appears to be a major breakthrough after months of threats and violence.
However analysts warn that such summits are usually the result of years of careful diplomacy so remain sceptical about what these rapidly arranged talks can achieve.
Mr Trump said the development was "great progress" but that sanctions will remain in place until a firm agreement is reached.
North Korea has not yet issued any official comment on the week"s developments.
South Korean National security adviser Chung Eui-yong, speaking outside the White House after the meeting, said: "I told President Trump that at our meeting North Korean leader Kim Jong-un said he"s committed to denuclearisation."
He added: "President Trump appreciated the briefing, and said he would meet Kim Jong-un by May to achieve permanent denuclearisation."
How did we reach this point?
North Korea has been isolated on the international stage for decades because of its well-documented human rights abuses and its pursuit of nuclear weapons, in defiance of international laws.
It says it needs nuclear weapons and missiles because its survival is under threat. It has carried out six nuclear tests, though it"s still unclear whether it could carry out a long-distance nuclear attack.
But South Korea"s hosting of the Winter Olympics gave an unexpected window for diplomacy, as rare inter-Korean talks were held to facilitate the North"s carefully choreographed attendance.
The South Korean delegation then held landmark talks with Mr Kim in Pyongyang earlier this week.
They returned with a statement saying the North was willing to denuclearise if it felt "it has no reason to retain nukes".
What has North Korea actually pledged?
There are four main things:
Mr Kim is prepared to sit down with the US president
It is "committed to denuclearisation"
It will halt all nuclear and missile tests
It understands that US-South Korean military drills "must continue".
The BBC"s Laura Bicker in Seoul says it is important to note that North Korea has not yet promised to abandon its nuclear weapons completely. It also remains unclear exactly what it is asking for in return.
The North has halted missile and nuclear tests during previous talks before, only to resume when it lost patience or felt it was not getting what it demanded.
The last point is also significant. The US has had tens of thousands of military personnel in South Korea since the end of the Korea War. The massive annual joint war games infuriate the North, because it believes they are preparation for invasion.
They were due to take place during the Olympics but have been suspended for now.
Is this a victory for Trump?
Mr Trump has repeatedly belittled Kim Jong-un, and last year threatened him with "fire and fury the likes of which the world has never seen before" if he continued to threaten the US. He has at times said there is no point in talking to North Korea.
But Mr Chung made a point of saying it was Mr Trump"s "maximum pressure policy" which had brought the parties to this point.
Christopher Hill, formerly the US negotiator with North Korea, told the BBC Mr Trump may have frightened Pyongyang by "looking at options that many American presidents would not have gone for".
But he has also shown flexibility to meet people other presidents might have dismissed, he says.
Our correspondent says Kim Jong-un has also scored a propaganda win, first with the Olympics and now by being seen to reach out to the US.
What about the other major players?
The South"s statement also credited "international solidarity" for reaching this point.
That is likely in part a reference to international sanctions, which have increased with each North Korean show of force.
China, North Korea"s main economic supporter, has in recent months toughened up its dealings with the North, including on key areas like petroleum and oil, which is thought to be putting a major strain on the North.
Japan, which saw North Korean missiles fly over its territory twice last year, is routinely distrustful of Pyongyang"s motives, saying it must commit to giving up its weapons before any talks can take place.
Prime Minister Shinzo Abe welcomed news of a Trump-Kim meeting, but said: "We will keep putting maximum pressure until North Korea takes concrete actions toward denuclearisation."
Have talks like this happened in the past?
No sitting US president has ever met a North Korean leader, but there have been repeated attempts to get North Korea to denuclearise.
The last major effort - the Six Party talks - collapsed in 2008, largely because North Korea refused to allow inspectors to verify that it had shut down its nuclear programme.
A number of bids to restart the talks also collapsed, including in 2012 when North Korea launched another rocket, two weeks after announcing a "leap day" agreement with the US.
Where might these talks happen?
Kim Jong-un is under a range of travel sanctions, so his options are limited. It is thought to be unlikely that Mr Trump would go to North Korea.
They could meet in the demilitarized zone - the border between the North and South - or more likely, China will host them.