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Trump, who is in Osaka, Japan, for a Group of 20 summit, is due to arrive in South Korea later on Saturday. He is scheduled to return to Washington on Sunday.
If Trump and Kim were to meet, it would be for the third time in just over a year, and four months since their second summit, in Vietnam, broke down with no progress on US efforts to press North Korea to give up its nuclear weapons.
Trump made the offer to meet Kim in a comment on Twitter about his trip to South Korea.
"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(?)!" he said.
Trump later told reporters his offer to Kim was a spur-of-the-moment idea: "I just thought of it this morning."
"We'll be there and I just put out a feeler because I don't know where he is right now. He may not be in North Korea," he said.
"If he's there, we'll see each other for two minutes, that's all we can, but that will be fine," he added. Trump said he and Kim "get along very well."
A senior North Korean official said a summit between Trump and Kim in the DMZ would be "meaningful" in advancing relations.
"We see it as a very interesting suggestion, but we have not received an official proposal in this regard," Choe Son Hui, North Korea's first vice-minister of Foreign Affairs said in a statement, North Korea's KCNA state media reported.
"I am of the view that if the DPRK-US summit meetings take place on the division line, as is intended by President Trump, it would serve as another meaningful occasion in further deepening the personal relations between the two leaders and advancing the bilateral relations," Choe said.
She was referring to North Korea by its official name - the Democratic People's Republic of Korea.
US special envoy Stephen Biegun said on Friday the United States was ready to hold constructive talks with North Korea to follow through on a denuclearisation agreement reached by the two countries last year, South Korea's foreign ministry said.
Biegun told his South Korean counterpart, Lee Do-hoon, that Washington wanted to make "simultaneous, parallel" progress on the agreement reached at a summit between Trump and Kim in Singapore in June last year, the ministry said in a statement.
But negotiations have stalled since their second summit in Vietnam in February collapsed as the two sides failed to narrow differences between US calls for denuclearisation and North Korean demands for sanctions relief.
South Korea's presidential office said nothing was confirmed with regards to a Trump, Kim meeting.
Russian President Vladimir Putin told South Korean President Moon Jae-in at a meeting on the sidelines of the G20 summit that Kim had told him in April security guarantees were key, and that corresponding measures were needed to realise denuclearisation, South Korea's presidential office said.
Trump wanted to visit the DMZ, on the border between North and South Korea, on a 2017 visit to South Korea but bad weather scuppered the plan.
He said before departing for the G20 summit that he did not expect to meet Kim on this trip.
US Secretary of State Mike Pompeo said this week that a recent exchange of letters between Trump and Kim boosted hopes for a restart of talks, calling it a "very real possibility."
North Korea's official KCNA news agency said Trump's letter had "excellent content" and Kim would "seriously contemplate" it, without elaborating.
Trump has previously said publicly he had received a very warm "beautiful letter" from Kim. He has not divulged its contents, but the White House official, who did not want to be identified, described the letter as "very flowery".
"President Trump made his pitch for a short summit with Chairman Kim on Twitter as White House officials most likely have tried - and failed - to set up such a meeting through official diplomatic or South Korean channels,"
Harry J. Kazianis of the Center for the National Interest said a short summit could set the scene for progress later.
"While no major agreements will be signed, both sides can reaffirm their commitment to dialogue and diplomacy, essentially resetting the table for a future deal in the weeks and months to come," Kazianis said.
Others were more sceptical.
"The fundamental problem - no working-level meetings and no basic change in at least the US negotiating position - means that any meeting right now is just pointless theatre," Vipin Narang, associate professor of political science at MIT, on Twitter.