However, Leong said both found it hard to hide their nervousness once they were seated, with Trump displaying a slanted smile, and fidgeting with his hands and Kim leaning and staring at the ground.
For his part, Trump appeared exceedingly optimistic about the ongoing talks.
The two leaders will issue a joint statement after the working lunch between Trump and Kim concludes.
"This is going to lead to more and more and more", Trump said. Korean Central News Agency (KCNA) mentioned that the summit is being held "under the great attention and expectation of the whole world".
Before they parted ways, he promised that the pair would meet many times and, when asked if he'd extend an invitation for his North Korean counterpart to visit the White House, he said: "Absolutely I will".
The US side however reportedly believes that an immediate tangible outcome is unlikely following the meet.
Pompeo stressed that the Trump administration would only accept complete denuclearisation of the North.. More news: US expands China health alert after acoustic incident
The two leaders exchanged broad smiles and warm thank-yous when they met at the Istana.
"It's hospitality that we would have offered them, and as Chairman Kim said yesterday, he would have liked to have come to Singapore anyway, with or without the summit", Dr Balakrishnan said in an interview with the BBC.
Trump was flanked in the larger meeting by chief of staff John Kelly, Secretary of State Mike Pompeo and national security adviser John Bolton.
State television aired the same photographs of Kim's departure as well as his arrival in Singapore and the North Korean leader's meeting with Singapore Prime Minister Lee Hsien Loong at the Presidential Palace. He lauded Kim as a "very smart, very good negotiator", saying the North Korean leader "might want to do this as much or more than me". "I too, got little sleep last night", Moon told his cabinet in Seoul as the summit began in Singapore.
Professor John Hajek from the University of Melbourne's School of Languages and Linguistics said the US President was "doing all the talking" and had the "upper hand" when the two met for their historic meeting.
Inside, they sat alongside each other against a backdrop of North Korean and U.S. flags, with Kim beaming broadly as the United States president gave him a thumbs up. "If diplomacy does not move in the right direction- and we are hopeful that it will continue to do so- those measures will increase".
Tuesday's statement, notably, contains no references to complete, verifiable, and irreversible denuclearization (CVID) - long a key USA demand going into talks with Pyongyang.