Differing reports stated that Kim was on an Air China aircraft, not a private jet.
Kim only started meeting with foreign leaders in recent months, slowly shedding a reputation as the reclusive ruler of the so-called Hermit Kingdom after summits with the Chinese and South Korean presidents.
While the summit reaffirms the U.S.'s close ties with Singapore - long one of America's best partners in South Asia - local officials have in recent weeks criticized Trump's efforts to reverse what he says are unfair trade practices.
Trump and others in his government have said advancement has been made on obtaining a commitment from Kim to give up all his nuclear weapons and long-range ballistic missiles.
But US Secretary of State Mike Pompeo said the two-hour meeting on Monday morning was "substantive and detailed".
As Trump and Lee sat down for a working lunch at the Istana house, Trump sounded optimistic, telling Lee, "we've got a very interesting meeting in particular tomorrow, and I think things can work out very nicely". The official was not authorized to speak publicly about internal deliberations and insisted on anonymity.
Observers in the region warn that ultimatums over denuclearization could make military confrontation between the US and North Korea even more likely, a situation Zhao Hai from China's Academy of Social Sciences described as a "down-spiral of security" that could turn into a nuclear "nightmare".
"It's never been done before", Trump said of a U.S.
Officials from USA and North Korean delegations have finished up a meeting in preparation of Tuesday's summit.
Even after touching down in Singapore, it was clear that the trade disputes with United States allies in Quebec continued to rankle Trump.
The North Korean autocrat's every move will be followed by 3,000 journalists who have converged on Singapore, and by gawkers around the world, up until he shakes hands with Trump on Tuesday.
In a post on Twitter during his flight, Trump reiterated his optimism that the summit with the North Korean leader will be a success.
South Korea's Yonhap news agency said the two leaders discussed denuclearization issues. He has not left his country since taking office in 2011 other than to visit China and the South Korean side of the Demilitarised Zone on the border.
Meanwhile, there was no indication of Kim's schedule for the whole of Monday, ahead of his meeting with Trump that begins at 9am on Tuesday. Trump has also raised the possibility of further summits and an agreement ending the Korean War by replacing the armistice signed in 1953 with a peace treaty.
Trump on Sunday called the unprecedented summit, at which North Korea's nuclear arsenal will be at the top of the agenda, a "one-time shot" at peace. Past nuclear deals have crumbled over North Korea's reluctance to open its doors to outsiders.
Another possibility from the summit is a deal to end the Korean War, which North Korea has long demanded, presumably, in part, to get USA troops off the Korean Peninsula and, eventually, pave the way for a North Korean-led unified Korea. Progress could lead to a peace treaty and formally end a state of war on the Korean Peninsula, which has persisted for almost 70 years. The North may see a treaty - and its presumed safety assurances from Washington - as its best way of preserving the Kim family dynasty.
Kim may be interested in getting aid and eventual investment to stabilize and then rebuild a crumbling economy.
Trump was pictured with the cake, delivered three days before his actual birthday.
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