It might be an awkward setting for Mr. Trump, though, who would be wary of not being seen as a supplicant. And North Korea might want to give Mr. Trump, who has expressed an interest in military parades, a display of its own.
When Secretary of State Madeleine K. Albright visited Pyongyang in 2000 in an attempt to convince Kim Jong-il, Kim Jong-un’s father, to halt his ballistic missile program, she attended a mass propaganda performance that included an image of the very missile she was trying to get North Korea to curtail.
Jeju Island, South Korea
The governor of the South Korean island of Jeju has proposed holding the meeting there. The island, south of the Korean Peninsula, is a tourist destination, and its relatively small size and population could make security easier than in a large city like Seoul, the South’s capital.
Washington would also be a potential spot, although Mr. Kim would most likely be wary of making the American capital his first trip abroad as North Korea’s leader.
A meeting there would also be awkward for the White House, which would be wary of the propaganda value it could give the North. When Marshal Jo Myong-rok, a high-level North Korean military official, visited Washington in 2000 to invite Mr. Clinton to Pyongyang, he first met with Dr. Albright while wearing a business suit. He then changed into a medal-festooned military uniform and high-brimmed hat to meet with Mr. Clinton, creating an uncomfortable image for the White House.
China is North Korea’s only significant ally, although their relationship has hardly been close in recent years. Still, China was one of the few countries Kim Jong-il traveled to as North Korea’s leader.
China has also played an active role in promoting negotiations among all sides and was a host to the so-called six-party talks a decade ago. Geng Shuang, a Chinese Foreign Ministry spokesman, said Friday that China welcomed the meeting and would “continue to make unremitting efforts” for a “peaceful settlement of the nuclear issue.” But he did not directly answer a question about whether Beijing would be host.
Geneva, the city in neutral Switzerland, has hosted high-level meetings between rivals, like between President Ronald Reagan and the Soviet leader Mikhail Gorbachev in 1985. And Kim Jong-un would have more familiarity with the country, where he studied in the late 1990s, than most other places.
Like China, Russia has been an occasional destination for North Korean leaders. Mr. Kim himself has not gone as leader, though. He canceled plans to travel to Moscow in 2015 for events to mark the 70th anniversary of the defeat of Nazi Germany in World War II. A visit to Moscow might not look good for Mr. Trump, either, given the charges that Russians tried to interfere in the 2016 election to help his campaign.
Sweden has long been a key intermediary between the United States and North Korea. The United States does not have an embassy in the North, and Sweden is the so-called protecting power that provides consular services for Americans, including meeting with citizens who are imprisoned there. Sweden has also been the site of talks between North Korean officials and experts from the United States, South Korea and elsewhere. And last week a Swedish newspaper, Dagens Nyheter, reported that Ri Yong-ho, the North Korean foreign minister, would visit Sweden soon, fueling speculation about a possible meeting location.
Ulan Bator, Mongolia
Mongolia, which shares borders with Russia and China, has pursued a policy of neutrality in recent years and has good relations with both the United States and North Korea. Tsakhiagiin Elbegdorj, a former Mongolian president, tweeted in support of a meeting in Ulan Bator, saying: “Here is an offer: US President Trump and NK leader Kim meet in UB. Mongolia is the most suitable, neutral territory.”
Because of an editing error, an earlier version of this article misstated the capital of Switzerland. It is Bern, not Geneva.