Kim has yet to publicly confirm his invitation to meet with Trump in a bid to defuse a standoff over Pyongyang's nuclear programme, and officials in Seoul and Washington say the exact location and timing of any summit remain to be determined.
But that hasn't stopped officials, analysts, and other observers from debating the pros and cons of possible summit sites, ranging from North Korea's capital city of Pyongyang, to the Joint Security Area (JSA) between the two Koreas, to farther afield in other areas of Asia or Europe.
Here are a few of the top locations being discussed:
JOINT SECURITY AREA, PANMUNJOM
One of the most likely sites being discussed is the Joint Security Area in Panmunjom that straddles the Military Demarcation Line between North and South Korea. It's the only spot along the heavily fortified Demilitarized Zone (DMZ) where North Korean troops stand face-to-face with South Korean and United Nations Command forces.
"Places like Switzerland, Sweden or Jeju Island have been gaining a lot of attention, but we also view the JSA as a serious option," an official with South Korea's presidential Blue House said on Sunday. Some sceptics see the JSA as a symbol of the Korean War and continued tensions, rather than a place for a peace deal.
However, it would allow Kim and Trump to meet without travelling far from either of their respective security forces or to be seen kowtowing in an "enemy" capital. Kim is scheduled to meet South Korean President Moon Jae-in at the JSA in April for their first ever summit, and South Korean officials see it as a top contender for a Trump summit as well.
"If North Korea and the U.S., who are the directly involved parties of the truce agreement, hold the summit at Panmunjom, it would hold the significant meaning of turning a symbol of division into one of peace," the Blue House official said.
Another spot in South Korea that has been raised as an option is scenic Jeju Island, off the southern coast, easily accessed by either boat or airplane from the Korean peninsula. Kim Jong Un's father and predecessor, Kim Jong Il, was known to be afraid of flying, but the younger Kim has been shown in official photos getting off airplanes and even "flying" a North Korean designed aircraft.
"In Korea, I'm sticking with Jeju Island. Jeju has soul, born of tragedy, and beauty and nature," said Yonsei University Professor John Delury, who led a discussion of possible summit locations on Twitter.The island hosts an annual "peace and prosperity" forum in May designed to attract international leaders."As the 'Island of Peace,' Jeju is the ideal place to hold the North Korea-US summit," Jeju's governor said in a statement.
Several spots in Europe, including Switzerland and Sweden, have been raised as possible neutral locations for a summit. Both countries have international reputations as mediators, and both played a role as members of the Neutral Nations Supervisory Commission helping to regulate relations between the two Koreas after the 1953 armistice that paused - but did not officially end - the Korean War.
As a child, Kim and his siblings attended an elite private school in Switzerland in the 1990s, according to former classmates there. However, Kim's time in the West is not an officially acknowledged part of his biography and analysts say he may want to avoid drawing attention to that period in socialist and impoverished North Korea. The Swiss foreign ministry has said the country is ready to facilitate talks between North Korea and the United States, and is in contact with both sides.
"It is up to the parties involved to decide if, when and where the talks will be held," the ministry said in a statement. Sweden's prime minister told reporters on Saturday that the country stands ready to help "in any way."
Sweden has held unofficial talks in the past, and North Korea's foreign minister is expected to visit soon, according to Swedish media.
A summit in Asia would be closer to home for the notoriously reclusive Kim, and cities including Beijing, Singapore, Hanoi, or even Ulaanbaatar in Mongolia have been raised as possibilities. Beijing is seen as one of Pyongyang's biggest backers, and has hosted a series of multilateral negotiations over North Korea held intermittently since 2003 and attended by China, Japan, North Korea, Russia, South Korea, and the United States.
In recent years, however, relations between Pyongyang and Beijing have cooled, and analysts say Trump may not be keen on providing a major rival like China a chance to steal some of the limelight. Mongolia's capital city, Ulaanbaatar, has been the site of past sensitive negotiations with North Koreans, maintains friendly relations with both sides, and has little in the way of political baggage.
Its lack of pizzazz, however, may not provide the backdrop Trump is seeking for his riskiest diplomatic gambit.
PYONGYANG OR WASHINGTON
As far as is publicly known, Kim has not left North Korea since he came to power in 2011, meaning any trip outside the country might be problematic.