Mr Trump had said on Saturday he would know within a minute of meeting Mr Kim whether he would reach a deal.
"I think that very quickly I'll know that something good is going to happen", Trump said in Canada where we met leaders of other G7 economies.
Starting with a handshake, President Donald Trump and North Korean leader Kim Jong Un made history Tuesday as they began their unprecedented summit aimed at staving off the threat of nuclear war.
Basketball star and improbable North Korean envoy Dennis Rodman broke down in tears live on television Tuesday as he recounted the hostility he faced for meeting dictator Kim Jong Un.
The world was watching as they smiled for the cameras, then sat down to make brief remarks before going into a almost 40-minute private meeting, with only translators present.
And with U.S. and North Korean officials reportedly struggling to close the gap between the two countries in pre-negotiations on Monday, giving up access to a full record of Kim's comments on Tuesday could hinder further talks.
When Singaporean Prime Minister Lee Hsien Loong visited the global 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".
For each leader, the meeting represents a major gamble, and its outcome will be dissected around the globe for a sense of whether one of the world's greatest security threats - Kim's nuclear arsenal - can be solved.
The US President is reportedly spending a bit more time with PM Lee today - two hours of talks and a working lunch together with government officials from respective countries, according to Channel NewsAsia.
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"I think things could work out very nicely", Trump said in a Tuesday meeting with Singapore's Prime Minister Lee Hsien Loong.
"This guy wants to be around the world, he wants to come to America, he wants to enjoy his life, he wants his people to enjoy this life", Rodman said, adding that Mr Kim is "not a dumb man".
The moment, as it played out, was slightly awkward, with Mr Kim seen staring at the ground and Mr Trump fidgeting with his hands.
The two leaders did not respond to reporters' questions on what they would be signing.
Trump says Kim has a "great personality" and is "very smart". President Trump committed to provide security guarantees to the DPRK, and Chairman Kim Jong Un reaffirmed his firm and unwavering commitment to complete denuclearization of the Korean Peninsula.
And until denuclearisation takes place, sanctions against North Korea would remain in place, he stressed.
The North Korea leader said through a translator that the document "heralds a new start, leaving the past behind".