With gaps remaining over what denuclearisation would entail, officials from both sides were trying to push the agenda forward ahead of Tuesday's summit.
A USA official says Monday that the leaders will meet for up to two hours before an expanded bilateral meeting with their respective advisers.
The American president tweeted Monday morning: "Great to be in Singapore, excitement in the air!"
Trump was joined at the meeting with Lee by USA officials including Secretary of State Mike Pompeo, White House Chief of Staff John Kelly and National Security Adviser John Bolton.
The U.S.is hoping to leave the summit with a commitment from North Korea to give up its nuclear weapons but it is not yet clear what it would take for North Korea to agree to such an agreement.
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"The entire world is focusing on the historic summit between the DPRK (North Korea) and the USA, and thanks to your honest efforts".
However, Reuters reported that Mr Kim would be leaving Singapore at 2pm tomorrow, citing a source involved in planning his visit.
"Wide-ranging and profound views on the issue of establishing new DPRK-U.S. relations, the issue of building a permanent and durable peace-keeping mechanism on the Korean Peninsula, the issue of realizing the denuclearization of the Korean peninsula and other issues of mutual concern, as required by the changed era, will be exchanged at the DPRK-U.S. summit talks", a dispatch by the North's official Korean Central News Agency (KCNA) said.
Trump and Kim arrived in Singapore hours apart Sunday.
The tests came amid a campaign of "maximum pressure", led by the United States, that tightened economic sanctions against North Korea and raised the possibility of military action.
Kim, who met Singaporean Prime Minister Lee Hsien Loong on Sunday after arriving, thanked him for hosting the event. "The scale is bigger, the number of journalists coming is huge, even more than what we had for the IMF (International Monetary Fund) and World Bank meetings (in Singapore in 2006)", he said.
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This will be the first summit of its kind between a leader of North Korea and a sitting US president.
Previous year the North carried out what was by far its most powerful nuclear test to date and launched a flurry of missiles - including two over Japan - while Trump threatened Pyongyang with "fire and fury" and Kim dubbed him a "mentally deranged USA dotard".
Another still photo in the news clip showed him shaking hands with North Korean officials, with another showing him waving from outside the Air China 747 jumbo which brought him to Singapore. The meeting was initially meant to rid North Korea of its nuclear weapons, but the talks have been portrayed by Trump in recent days more as a get-to-know-you session.
South Korean president Moon Jae-in was not expected to arrive in Singapore on Tuesday, which also lowered expectations about whether a comprehensive peace treaty to end the Korean War would be formally signed.
The jet landed at the airport Sunday afternoon amid huge security precautions on the city-state island.
It is the farthest Kim has travelled since inheriting power in 2011, and only his third known trip outside the country since then, with the use of a Chinese plane raising questions over the state of North Korea's ageing fleet of Soviet-built aircraft.
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