Trump was joined in the talks by Secretary of State Mike Pompeo, who played a crucial role setting up the summit, National Security Adviser John Bolton and Chief of Staff John Kelly. "We're going to have a great discussion and I think ... it will be tremendously successful", he said.
However, it now appears that the U.S.is confident tings are going smoothly. They walked outside together after the lunch, stopping briefly to look at the US president's special limousine.
Tuesday marked the first meeting between a sitting USA president and a North Korean leader.
Korean-Americans make a toast at a party to watch the Trump-Kim Singapore summit in Koreatown, Los Angeles.More news: Neymar on target again as Brazil beat Austria
For each leader, the summit represented a major gamble, and the 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 eliminated.
"It was not easy to get here", he said.
While the president has more recently sought to temper expectations of what he may be able to achieve with the summit, he has also said the interaction amounts to a crucial test in determining whether Kim is genuine in committing to rid his country of its nuclear program.
"There will be challenges ahead", Kim said, but he vowed to work with Trump.More news: Donald Trump, Kim Jong-un arrive in Singapore for historic summit
US and North Korean officials huddled throughout Monday at the Ritz-Carlton hotel, outlining specific goals for what the leaders should try to accomplish and multiple scenarios for resolving key issues, a senior U.S official said, adding that the meetings were also an ice breaker of sorts, allowing the teams to get better acquainted after decades of minimal contact between their nations.
Singapore's Ministry of Foreign Affairs (MFA) released a statement saying the two discussed mutual relations and the situation on the Korean peninsula, with Prime Minister Lee thanking Kim for his and Trump's "bold and admirable decision" to hold the summit.
"Trump seemed to be very aware of this, that he needed to up the stakes and be seen that he is the leader".
Setting out the United States position before the summit, Pompeo stressed that the Trump administration would only accept complete denuclearisation of the North. He was was sent home from North Korea in a coma and died soon after.
There will not be a repeat of "flimsy agreements" made between previous USA administrations and North Korea, Secretary Pompeo told reporters in Singapore on Monday.More news: Three Saudi civilians killed by Houthi-launched missile in Jazan
But in return, Washington would offer "different and unique" guarantees "to provide them sufficient certainty that they can be comfortable that denuclearisation is not something that ends badly for them".