The summit has drawn thousands of journalists to the Asia city-state. "Meeting the leader of the free world is automatically legitimacy branding".
A jet carrying Kim landed hours earlier.
Against a backdrop of US and North Korean flags, the two first shook hands for about 12 seconds, and Trump at one point grabbed before leading him toward another room where they are scheduled to meet one-on-one, with just translators present, for about 45 minutes.
"I just think it's going to work out very nicely", he said during a lunch with Singapore's prime minister on Monday.
AFP 2018 Saul Loeb World Follows Every Step of Historic Trump Kim Summit
"But we overcame all of them and we are here today".
Trump is set to meet with Lee on Monday. The official was not authorized to speak publicly about internal deliberations and insisted on anonymity.
The historic first meeting between a sitting US president and and North Korea's leader will take place Tuesday at 9 a.m. when Trump and Kin will meet alone with only translators in the room.
Trump wants the complete, verifiable and irreversible dismantling of North Korea's nuclear weapons program.
Entering their meeting, Trump said that he "feels really great".More news: 'Ocean's 8' is No. 1 at the Box Office
"Thanks PM Lee for spending $20 million of taxpayers money, which can. help a lot of needy families in Singapore to survive", posted one Facebook user.
The real North Korean leader and Trump arrived in Singapore on Sunday.
Trump patted Kim on the back and later placed his hand on the North Korean's shoulder as they walked into one-on-one talks.
Kim has shown surprising diplomatic skills in his meetings with overseas leaders this year.
U.S. officials and the North Korean state news agency struck a hopeful tone as the meeting approached, while negotiators from both countries met privately in the last hours to iron out agreements, such as over the definition of denuclearisation and the terms under which Pyongyang might agree to dismantle its nuclear arsenal. It heralded the summit as part of a "changed era".More news: Singapore summit: Will striking a deal with Trump risk Kim's regime?
But he has since lowered expectations, saying the talks would be more about starting a relationship with Kim for a negotiating process that would take more than one summit. While Kim is staying at the St Regis Hotel that has been converted into a fortress ahead of June 12, Trump is reportedly staying at Shangri-La Hotel.
Officials of the two sides held last-minute talks to lay the groundwork for the summit of the old foes, an event nearly unthinkable just months ago, when they were exchanging insults and threats that raised fears of war.
At the conclusion of the summit, President Trump will participate in a media availability before departing tomorrow at approximately 8 p.m. for the United States. China and South Korea would have to sign off on any legal treaty. And he made clear that, whatever the role of diplomats and aides in setting up the meeting, it would be the personal exchange between him and Kim that would be decisive. Pyongyang's weapons program benefits from forced labor.
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. But the North has long sought an end to the U.S. military presence in the South, where Washington has around 28,000 troops stationed to protect it from its neighbour.
Cheering crowds greeted Kim during his stroll, and cameras from North Korea's state-run Korean Central News Agency documented his every move.More news: Miyazaki is directing Sekiro: Shadows Die Twice and it's out next year