Mr Kim said to Mr Lee: "The entire world is focusing on the historic summit between the DPRK (Democratic People's Republic of Korea) and the USA, and thanks to your honest efforts. we were able to complete the preparations for the historic summit, and I would like to thank you for that".
Critics worry Kim has already won by scoring such a high-profile meeting..
However, last week, Trump re-instated the summit after meeting with Kim's top aide, Central Committee of the Workers' Party of Korea Vice Chairman Kim Yong Chol, at the White House. Pyongyang says it hopes for a "new" start with the US. The official was not authorized to comment publicly about on internal deliberations and requested anonymity. The plan is to wrap up the meeting in the evening and, if things go well, for there to be a joint declaration, according to another USA official.
Mr Lee also said it was "not easy" to find a location in the world that is suitable to host such a high-level meeting.
Trump and Kim are set to meet at 9 a.m. Tuesday in Singapore, which is 9 p.m. Monday on the U.S. East Coast.
Just meeting with Mr Trump will also give Mr Kim recognition as the leader of a "normal" country and as an equal of the U.S. leader.
Authorities have beefed up security around the hotels where Trump and Kim (Saint Regis) will be staying. They urged Trump to maintain a tough stance with Kim and his ally China, writing that any deal giving North Korea relief on sanctions without the ability to verify that it is dismantling its nuclear and missile arsenal "is a bad deal".
The 22-year-old American student was detained in North Korea for 17 months and was returned to his family "with severe brain damage and in a nonresponsive state" last June.
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North Korea's leader Kim Jong Un shakes hands with Singapore's Prime Minister Lee Hsien Loong at the Istana in Singapore on June 10.
Part of the interest in Tuesday's summit is simply because Kim has had such limited appearances on the world stage.
His trip to Singapore is the furthest he has traveled from his home country since assuming the mantle of Supreme Leader.
Trump initially touted the potential for a grand bargain with North Korea to rid itself of a nuclear missile program that has advanced rapidly to threaten the United States. "Donald Trump is not going to capitulate, so there's really only two options- peace or war".
But the South's Winter Olympics in February were the catalyst for a flurry of diplomatic moves as South Korea's dovish leader Moon Jae-in sought to bring the two sides together.
It's unclear what Trump and Kim might decide Tuesday.
Any nuclear deal will hinge on North Korea's willingness to allow unfettered outside inspections of the country's warheads and nuclear fuel, much of which is likely kept in a vast complex of underground facilities. This has included backing away from an original demand for North Korea's swift denuclearisation.
The North invaded the South in 1950 and the ensuing war pitted US-led United Nations troops backing Seoul against Pyongyang's forces which were aided by China. "That includes inspecting the North Korean nuclear shutdown, as well as a role in the economic development Kim is seeking for his country".