The U.S. President earlier promised that "working together ... we will get it done".
With less than 24 hours until his historic face-to-face encounter with Kim, Trump spent his Monday in Singapore fuming on Twitter over his weekend trade spat with G-7 allies and paying a visit to Singapore's prime minister.
Mr Kim - who is the Chairman of the State Affairs Commission, North Korea's highest decision-making body - also shook hands with Dr Vivian, Mr Ong, Senior Minister of State for Foreign Affairs as well as Defence Maliki Osman and Senior Minister of State for Communications and Information as well as Culture, Community and Youth Sim Ann.
The country has diplomatic ties with both North Korea and the US.
He also made a stop at the worldwide media centre at the F1 Pit Building, which will be the base for the thousands of journalists in town to cover the meeting between Mr Trump and Mr Kim.
Joining Trump were Secretary of State Mike Pompeo, Chief of Staff John Kelly, National Security Advisor John Bolton, U.S. Ambassador to the Philippines Sung Kim, press secretary Sarah Huckabee Sanders, and Deputy Assistant to the President for Asian Affairs Matthew Pottinger.
Mr Lee is slated to meet Mr Trump today at the Istana.
Last week, Trump told reporters that it would be easy to tell if a deal was possible with Kim in the first minute of the meeting.
Secretary of State Mike Pompeo, meanwhile, said on Monday that preparatory talks had moved "quite rapidly", with the US prepared to provide North Korea security assurances that are "unique" from what it has previously offered.
Trump has also raised the possibility of further summits and an agreement ending the Korean War by replacing the armistice signed in 1953 with a peace treaty.
Before the summit meeting, Trump called the Japanese prime minister, Shinzo Abe, and the South Korean president, Moon Jae-in, to brief them on developments.
Asked if he would meet Mr Kim in Singapore, he said he thought the North Korean leader had "bigger things to worry about", but added that every time he had met with Mr Kim, it had been a surprise. "And it's my honour and we will have a terrific relationship". But there are major doubts, given how hard it has been for Kim to build his program and given that the weapons are seen as the major guarantee to his holding onto unchecked power.
The smiles on Tuesday "don't mean anything" unless they are followed by specific commitments to make change in North Korea, Mr. Pinkston said.