In a Saturday morning tweet from his hotel in Japan, Trump said if Kim was interested he'd be open to a greeting on the border.
In response, state news agency KCNA quoted a senior North Korean official several hours later saying it was a "very interesting suggestion" and would be a "meaningful occasion", but that North Korea had not received an official proposal.
Trump confirmed to reporters Saturday that the United States had heard from North Korea and had extended an official invitation.
Last week, Mr Trump - who this month said North Korea under Mr Kim's leadership had "tremendous potential" - sent the North Korean leader a personal letter whose content Mr Kim praised as "excellent".
Trump wanted to visit the DMZ during a 2017 visit to South Korea, but heavy fog prevented it. Kim and Moon held their historic first summit in the zone previous year, which preceded the U.S.
President Trump needs to mend fences with China after becoming entangled in a standoff with Iran that can turn sour any minute. US officials said sanctions on North Korea would stay in place until North Korea takes firmer steps toward nuclear disarmament.
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When asked if he would step across the border into North Korea, President Trump responded, "Sure I would". "If Trump and Kim meet and can announce some kind of interim agreement, that's great". Conservatives generally support a more confrontational approach to North Korea, while liberals advocate more engagement.
Yonsei University Professor John Delury tweeted, "To get real progress toward denuclearization-not to mention lots of other things that matter-North Korea has to be made to feel less threatened, more secure, less under siege, more welcome in the world..."
"I don't understand why the president would want to meet with a person who has not changed a single bit", Gardner said. The two leaders then took a stroll around the office complex and shared cocktail drinks together before holding a dinner meet at Sangchunjae pavilion, a attractive structure modelled after a traditional Korean house, according to the news report.
Mr Trump made curtailing illegal immigration and building a wall on the US-Mexico border a central theme of his 2016 campaign, though he has struggled to fulfil his pledge.
He said before departing for the G20 summit that he did not expect to meet Kim on this trip.
Both sides had agreed to establish new relations and work towards denuclearisation of the Korean peninsula. According to a Blue House spokesperson, Putin spoke of Kim's emphasis on the importance of North Korea's security issues in the negotiations... and that there needs to be corresponding measures in exchange for denuclearization. "If in the end they don't meet, it's good that Trump offered to".
Trump is scheduled to depart Osaka on Saturday evening for Seoul, where he will spend one night before returning to Washington.