Trump on Monday forecast a "nice" outcome, while Kim spent the day out of view.
Few people expect Kim to commit to what some Trump administration officials call "CVID:" Complete, verifiable, and irreversible dismantlement of nuclear weapons programs - certainly not at this meeting, and perhaps not ever.
During the media interview, Mr Lee said that Singapore had committed a lot of resources to the summit's logistics and security because of the profile of the meeting and the nature of what is being discussed. He said the summit will lay the ground work for the "hard work" ahead.
Trump landed in the evening after a long flight from Canada and the Group of Seven meeting there, telling Singaporean officials who welcomed him that he was feeling "very good" about the summit.
Preliminary meetings have already begun.
Along with an extensive security detail, the North Korean leader took his own personal toilet to the meeting.
Maybe that is why they are living it up.without paying the bill.
"The Police seek the understanding and cooperation of the public as these measures are necessary to ensuring the security of the summit", it added.
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Mr Kim, who is the chairman of the State Affairs Commission of North Korea, headed to the Istana on Sunday where he met Mr Lee for the first time. The leaders are spending the night at hotels about a half-mile apart in the upscale shopping area in Tanglin.
North Korean leader Kim Jong Un watches the launch of a Hwasong-12 missile in this undated photo released by North Korea's Korean Central News Agency (KCNA) on September 16, 2017. However, others reported that the Air China Boeing 747 took off before the leader's jet did by an hour or two as a security measure to avoid making his mode of transportation known.
Trump and other top US officials have promised a "bright future" for the North if it quickly relinquishes its nuclear weapons - including economic inducements and a security agreement. The president also refused to sign a G-7 joint statement pledging cooperation.
It is no easy task. At the same time, Trump stressed that Washington would not accept Pyongyang's refusal to denuclearize.
Larry Kudlow, Trump's economic adviser, said Sunday on CNN that Trudeau's criticism had damaged Trump's prestige before the summit - and that's why Trump reacted so strongly.
He said Iran views Mr Trump and the United States with "great pessimism", saying they are known for "quitting treaties and violating their commitments".