Trump made the offer on Twitter on Saturday (Japan time), hours before he was due to visit Seoul from Osaka, Japan, where he has been attending a Group of 20 summit.
"After some very important meetings, including my meeting with President Xi of China, I will be leaving Japan for South Korea (with President Moon)," Trump wrote.
"While there, if Chairman Kim of North Korea sees this, I would meet him at the Border/DMZ just to shake his hand and say Hello(?)!"
|(Captured from US President Donald Trump's Twitter account)|
There had been speculation that a Trump-Kim meeting could happen inside the Demilitarized Zone when he visits Seoul on Saturday and Sunday.
But Trump denied before leaving Washington on Wednesday that he planned to meet Kim, saying only that he may speak to the North Korean leader "in a different form."
At a meeting in Osaka with Saudi Arabia's crown prince, Trump confirmed that he will be going to the DMZ. That part of the trip had previously been reported to be under consideration.
"We're going there," he said, according to news reports. "If he's there, we'll see each other for two minutes," he added, noting that he's unsure whether Kim is in the country.
If the two meet, it will be their third time seeing each other after their historic first summit in Singapore last June and their second summit in Vietnam in February.
The US and North Korea have been negotiating the dismantlement of Pyongyang's nuclear weapons program in exchange for sanctions relief, but talks have stalled since the second summit ended without a deal.
An exchange of personal letters between Trump and Kim this month led to hopes for a revival of talks.
Stephen Biegun, the US point man on the North, met with his South Korean counterpart, Lee Do-hoon, in Seoul on Friday and expressed the US' willingness to hold "constructive" dialogue with the North, according to South Korea's foreign ministry.
Biegun also said the US was ready to advance the commitments made at the Singapore summit "in a simultaneous and parallel manner."
The first Trump-Kim meeting yielded an agreement to establish "new" relations between the two countries and build a lasting peace regime on the Korean Peninsula. It also committed the North to work toward complete denuclearization in exchange for US security guarantees.
South Korean President Moon Jae-in and Trump are expected to discuss ways to resume the negotiations. Moon has been eager to act as a mediator between North Korea and the US, and held three summits of his own with Kim last year.
The South Korean president said in a written interview with Yonhap News Agency and six other global news agencies this week that the US and the North have been engaged in back-channel talks over a third summit.
The North later denied that such discussions were taking place.
It will be Trump's second visit to South Korea since taking office. The first was in November 2017. Trump tried to go to the DMZ during that trip too, but heavy fog forced his helicopter to