Trump said Friday that the list of potential sites for the meeting had been narrowed to two, without saying where. Some reports have said that Singapore, a stable and prosperous financial hub in Southeast Asia, is one of the potential locations.
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The meeting, expected in the coming weeks, follows a dizzying detente in recent months between the nuclear-armed North and its neighbour, which resulted in Friday's historic summit between Kim and the South's President Moon Jae-in.
The leaders agreed to pursue a permanent peace treaty and the complete denuclearisation of their divided peninsula.
Speaking at a meeting of Southeast Asian leaders, Singapore's Prime Minister Lee Hsien Loong dampened speculation about the city-state potentially hosting the Trump-Kim talks.
"As for the venue, we've also read the same reports as you in the newspapers about the possible places where a US-North Korea meeting can take place," he told reporters.
"We've had no formal invitations, requests from any of the parties. It has to be something agreed by both North Korea as well as the United States.
"I doubt very much they have come to any landing yet."
A diplomatic source, who spoke on condition of anonymity, said that he was not aware of any request, even informally.
Speculation has been rife about where the US and North Korean leaders will meet.
Sweden, Finland, Switzerland, and the capitals of China and Mongolia have been mentioned in reports, in addition to Singapore and the Demilitarized Zone that divides the two Koreas.
Landmark summits have taken place Singapore in the past -- the presidents of China and Taiwan met for the first time ever in the city-state three years ago after decades of estrangement following a traumatic split at the end of a civil war in 1949.
China's Xi Jinping and Taiwan's Ma Ying-jeou shook hands for more than a minute and smiled for a mass of reporters before holding talks.
Singapore and Pyongyang have diplomatic ties and the North has an embassy in the city-state.
In November, Singapore suspended trade ties with the North in the latest move by a country to implement UN sanctions to curb Pyongyang's nuclear weapons programme.
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