The US leader is due to visit South Korea after the G20 summit in Japan.
He arrives in Seoul on Saturday for a two-day trip aimed at rescuing foundering denuclearisation talks with North Korea.
In an apparently spontaneous gesture, Mr Trump unexpectedly mooted a meeting with Mr Kim.
Speaking at the G20 summit in Osaka, Mr Trump gave a few more details, telling reporters he decided on Saturday morning to "put out a feeler" to Mr Kim.
"If he's there, we'll see each other for two minutes and that's fine," he said in comments to reporters. He was speaking at the start of a working breakfast with Saudi Arabia's Crown Prince Mohammad Bin Salman.
He made the comments ahead of key talks with Chinese President Xi Jinping over a tit-for-tat trade war seen as a drag on global growth.
It remains unclear whether officials with Mr Trump were briefed in advance about his overture to the North Korean leader.
Mr Trump attempted to make a surprise visit to the demilitarised zone (DMZ) separating the two Koreas in November 2017, but was forced to abandon the plans due to bad weather.
How are North Korea-US relations?
They've soured somewhat since Mr Trump and Mr Kim met in Hanoi, Vietnam in February.
The summit - their second after Singapore in June 2018 - ended abruptly without agreement on North Korea's progress towards denuclearisation, with Mr Trump walking away from what he saw as a "bad deal".
Mr Trump has repeatedly insisted that North Korea must dismantle its nuclear arsenal before economic sanctions can be lifted.
Since the Hanoi summit, North Korea has risked incurring the wrath of the Trump administration by testing several short-range missiles.
But Mr Trump, who once threatened North Korea with "fire and fury", has struck a more conciliatory tone in recent months, speaking warmly about Mr Kim.
Last week, he sent the North Korean leader a personal letter, whose content Mr Kim praised as "excellent".
Earlier this month, he told reporters that North Korea under Mr Kim's leadership had "tremendous potential".
And in May during a visit to Japan, Mr Trump described Mr Kim as a "very smart guy" and said he expected "a lot of good things" to come out of North Korea.
Will Mr Trump actually meet Mr Kim?
Given the apparent spontaneity of Mr Trump's tweet, the prospect of him meeting Mr Kim at the DMZ seems doubtful.
His previous meetings with Mr Kim were planned well in advance.
But last week, a South Korean official said Mr Trump was considering a trip to the DMZ, prompting speculation a meeting with Mr Kim could be possible.
US and South Korean officials have sought to play down a potential rendezvous between the two leaders.
Media captionDonald Trump and Kim Jong-un: From enemies to frenemies
In contrast, Mr Trump has alluded to possible encounters, including "speaking to him (Mr Kim) in a different form" and seeing each other "for two minutes".
Instead, the pair may speak over the phone or write to each other another letter, Vipin Narang, security studies professor at the Massachusetts Institute of Technology, told the Telegraph.
"I wouldn't expect a Trump-Kim meeting: it's too risky to do so with such little working level prep, since neither can afford a second failed meeting at home," he said.
What is Mr Trump expected to discuss in Seoul?
The South Korean president's office has said Moon Jae-in and Mr Trump will discuss "how to closely co-operate with each other to build lasting peace through complete denuclearisation of the Korean peninsula".
Mr Trump and Mr Moon are both keen to bring Kim Jong-un back to the negotiating table
The US's special envoy on North Korea, Stephen Biegun, has been in Seoul recently for preparatory talks.
He said on Friday, in comments released through South Korea's foreign ministry, that the US was "ready to hold constructive discussions with North Korea".
Moon Jae-in: Caught between Trump and Kim
Mr Moon, who has had two genial meetings with Mr Kim, has made resolving the North Korean issue the focal point of his presidency.
He was mediator for the first Trump-Kim summit and is keen to get talks back on track, though was accused by North Korean state media this week of "meddling" in the process
Mr Trump will also be keen to demonstrate concrete progress as campaigning gets under way for the 2020 US elections.
The future of the US-South Korean post-war security alliance is also likely be on the agenda in Seoul.
What are the US-South Korea war games?
The US has about 30,000 personnel based in South Korea. Mr Trump has repeatedly said he would like to scale back this deployment.
The two countries' militaries conduct annual war games which infuriate North Korea, but they have scaled them back since ties warmed last year.
What do the other countries involved want?
North Korea has been a significant topic of conversation on the sidelines of the G20 summit in Osaka this week.
All involved parties agree that it would be better if there were no looming nuclear threat on the Korean peninsula, but differ in their views on how this should be brought about.