Mr Kim and his delegation met Mr Lee at the Istana shortly after arriving in Singapore on Sunday. "The entire world is focusing on the historic summit, thanks to your sincere efforts," the North Korean leader said through an interpreter.
Sunday's meeting marked the first time Mr Lee and Mr Kim have met, according to Singapore's Ministry of Foreign Affairs (MFA).
Mr Lee in turn thanked Mr Kim for making the rare foreign trip and said Singapore has been following developments on the Korean peninsula for a long time.
Mr Kim arrived in Singapore on Sunday afternoon (June 10) amid a media frenzy and heightened security. He is due to meet Mr Trump on Tuesday to discuss North Korea's nuclear programme.
The US President also flew to Singapore from Canada, where he participated in an especially contentious G7 summit that came after he triggered a trade dispute with several of the US' closest allies.
The two leaders are to meet on Tuesday at the Capella Hotel on Sentosa island where talks on denuclearisation of North Korea in exchange for economic help will be on the table.
Trump posted messages on Twitter on his way to Singapore, where he will meet Kim on Tuesday.
Trump said the summit with Kim is "a chance to achieve a truly wonderful result for North Korea and the world."
He writes "it will certainly be an exciting day" and he knows that Kim "will work very hard to create peace and prosperity for his land".
Trump tweeted on Sunday: "That is something that has rarely been done before."
The president added he has a feeling that what he calls the one-time opportunity "will not be wasted."
