But after a flurry of diplomatic overtures eased tension in recent months, the two leaders are now headed for a history-making handshake that USA officials hope could eventually lead to the dismantling of a North Korean nuclear programme that threatens the United States. Secretary of State Mike Pompeo said the delegations were "rapidly" making progress, but it will come down to the meeting between the two leaders.
The White House said later that discussions with North Korea had moved "more quickly than expected" and Trump would leave Singapore on Tuesday night after the summit, rather than Wednesday, as scheduled earlier. Only after their tête à tête will the President's top advisers - Pompeo, chief of staff John Kelly and national security adviser John Bolton - join him for an expanded bilateral meeting and working lunch.
On the eve of the summit, Trump stayed in his hotel but Kim made a decision to take a sightseeing tour of Singapore's hot spots, visiting botanical gardens at Gardens by the Bay and posing for a selfie before making a stop at Singapore's iconic three-tower Marina Bay Sands hotel.
He said on Saturday that he thought that he would know "within the first minute" of the meeting how serious Kim would be to give up its nuclear program.
What is unclear whether the summit will lead to any type of substantive outline of an agreement in which North Korea will halt its program and agree to denuclearize, and what kind of verification procedure it would accept.
It was a far cry from previous year when Trump threatened North Korea with "fire and fury" and mocked Kim as "little rocket man", and Kim denounced the US President as the "mentally deranged US dotard". It will certainly be an exciting day and I know that Kim Jong-un will work very hard to do something that has rarely been done before ...
Trump was presented with a birthday cake by his Singaporean host.
Pompeo also said that Trump had called Japanese Prime Minister Shinzo Abe and South Korean president Moon Jae-in. According to the Chicago Sun-Times, he told reporters that he wasn't sure if he would even meet Kim in Singapore.
Trump and Kim arrived not long ago on Singapore's Sentosa Island, the site of their unprecedented summit.
Any nuclear deal will hinge on North Korea's willingness to allow unfettered outside inspections of the country's warheads and nuclear fuel, much of which is likely kept in a vast complex of underground facilities.
When Singaporean Prime Minister Lee Hsien Loong visited the worldwide media center for the summit at F1 Pit Building on Sunday, he said it will cost the city state "around 20 million Singaporean dollars" (W16.1 billion, US$1=W1,078) to host the "historic summit".
With destiny calling, Trump was the first to arrive at the colonial-era Capella Hotel on Sentosa, a resort island off Singapore's port with luxury hotels, a Universal Studios theme park and man-made beaches.
North Korea signed onto a goal of "complete denuclearization" of the peninsula in an April 27 declaration with South Korea.
"There is no chance to make North Korean leaders. surrender their nuclear weapons", Andrei Lankov, a Korea expert at Seoul's Kookmin University, wrote recently on a Washington-based Asia newsletter.
"The entire world is watching the historic summit between (North Korea) and the United States of America, and thanks to your honest efforts ... we were able to complete the preparations for the historic summit", Kim told Lee through an interpreter.