The unprecedented meeting at the demilitarised zone (DMZ) comes after Mr Trump tweeted what he said was a spontaneous invitation to Mr Kim.
South Korea’s president said they would have a “handshake for peace”.
The North has not yet commented. The expected meeting raises hopes for a revival of stalled nuclear talks.
It will be the third meeting between Mr Trump and Mr Kim in just over a year, and their first since a summit in Vietnam broke down in February.
Reacting to Mr Trump’s offer on Saturday, North Korea said it was “very interesting”.
No sitting US president has ever been inside the DMZ, the thin strip of land which divides North and South Korea.
Mr Trump visited a guard post on the South Korean side of the DMZ alongside South Korea’s President Moon Jae-in. His meeting with Mr Kim is expected to take place at the Joint Security Area, where all negotiations between the South and North are held.
In a press conference in Seoul earlier, Mr Trump said he and Mr Kim had “developed a very good relationship” and he was looking forward to meeting him.
“It’s going to be very short, virtually a handshake. But that’s OK. A handshake means a lot.”
Apparently confirming that the invitation was not long planned, he said he arrived in South Korea and thought: “Hey, I’m here let’s see whether I can say hello.”
Speaking alongside Mr Trump, Mr Moon said the leaders would meet in Panmunjom, the so-called truce village inside the DMZ, where negotiations between South and North Korea have often taken place.
Mr Moon said the timing of a third full summit “depends on what change today’s meeting and dialogue could generate”, adding that “peace takes more courage compared to tensions”.
Why will this meeting be significant?
With no time for the all-important backroom diplomacy, it is expected to be largely a photo opportunity. However, it will be seen as a sign of their ongoing commitment to the denuclearisation talks.
Negotiations with North Korea, to try to convince it to abandon its controversial nuclear programme, reached a peak last year when Mr Trump and Mr Kim had a historic meeting in Singapore.
They both committed to the “complete denuclearisation” of the Korean peninsula, but without clarifying what that meant.
It was hoped their second meeting, in Hanoi in February, would make some concrete agreement about North Korea handing over its nuclear programme in exchange for some of the tight sanctions against it being lifted.
But those talks ended with no deal, as they failed to agree on the pace at which sanctions should be eased. Since then the negotiations have stalled, though Mr Kim and Mr Trump have exchanged letters recently.
What is the DMZ?
The DMZ, which runs about 4km (2.5 miles) wide and 250km long, has divided the peninsula since the Korean War ended in 1953.
Though that area, by definition, has no military installations or personnel, beyond it lies one of the most heavily militarised borders in the world.
The Joint Security Area (JSA) located at the Panmunjom village straddles the Military Demarcation Line and is where all negotiations between the two countries are held.
Tourists can also go to the JSA when relations between the two countries – still technically at war – allow it. No US sitting US president has been inside it. Bill Clinton once described it as the “scariest place on Earth”.