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.
The official media has reported that the two leaders plan to meet, but has offered few specifics, including where and when.
Olivia Enos, policy analyst with the Asian Studies Center at The Heritage Foundation in Washington, said "a successful summit would move beyond mere statements about denuclearisation".
It wasn't immediately clear what specific progress, if any, had been made in preliminary discussions between United States and North Korean officials in the run-up to the Tuesday summit.
Preliminary meetings have already begun.
Here is a rundown of the biggest actions that took place overnight, as Singapore is 12 hours ahead of Eastern Daylight Time. It also annoyed Singapore residents by offering to pay Kim's hotel 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. The leaders are spending the night at hotels about a half-mile apart in the upscale shopping area in Tanglin.
Trump hopes the landmark talks will kick-start a process that eventually see Kim give up his nuclear weapons in an irreversible manner that can be verified internationally.
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The two leaders have had an extraordinary up-and-down relationship over the past 18 months. It will be the first time a sitting USA president has met with a North Korean leader.
He had his first formal meeting of the summit with the prime minister of Singapore, Lee Hsien Loong.
It is unclear, however, under what terms North Korea would agree to in an effort to reduce its nuclear arsenal.
It is no easy task. Trump and Kim Jong-Un have both suggested they could also pursue a peace treaty to officially end the Korean War, which ceased because of an armistice signed in 1953. "And he has that opportunity, and he won't have that opportunity again", Trump said.
Trump told reporters in Canada on Saturday any agreement with Kim would be "spur of the moment", underscoring the uncertain outcome of what he called a "mission of peace".
"No", he said, "I don't feel unsafe". "Just my touch, my feel".
"In this case, what is important is that the summit is being held, and we are hosting it, not extravagantly but with due consideration to costs and making sure operational requirements are met", he said.
"It's unknown territory, in the truest sense", Trump said during the G-7.
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