While Trump was optimistic about prospects for the first-ever meeting of sitting U.S. and North Korean leaders, U.S. Secretary of State Mike Pompeo injected a note of caution, saying it remained to be seen if Kim was sincere about his willingness to denuclearise.
Officials of the two sides held last-minute talks to lay the groundwork for the summit of the old foes, an event almost unthinkable just months ago, when they were exchanging insults and threats that raised fears of war.
The summit was due to begin at 9 a.m. (0100 GMT)
Trump was the first to arrive at the Capella Hotel on Sentosa, a resort island off Singapore's port with luxury hotels, a Universal Studios theme park and man-made beaches. island, a resort. Television images showed Kim's motorcade arriving later.
Staff-level meetings between the United States and North Korea were going "well and quickly," Trump said in a message on Twitter on Tuesday.
But he added: "In the end, that doesn't matter. We will all know soon whether or not a real deal, unlike those of the past, can happen!"
The combatants of the 1950-53 Korean War are technically still at war, as the conflict, in which millions of people died, was concluded only with a truce.
On Tuesday morning, Pompeo fed the mounting anticipation of diplomatic breakthrough: "We're ready for today."
Trump will hold a one-on-one meeting with Kim on the resort island of Sentosa, before they are joined by officials and have lunch together, the White House has said.
While Trump and Kim shake hands, searching each other's eyes and words for signs of trust or deceit, the rest of the world will be watching, hoping that somehow these two unpredictable leaders can find a way to defuse one of the planet's most dangerous flashpoints.
"I just think it's going to work out very nicely," Trump said on Monday, although gaps remain over what denuclearisation would entail.
Pompeo told reporters the event should set the framework for "the hard work that will follow", insisting that North Korea had to move toward complete, verifiable and irreversible denuclearisation.
North Korea, however, has shown little appetite for surrendering nuclear weapons it considers vital to the survival of Kim's dynastic rule.
Sanctions on North Korea would remain in place until that happened, Pompeo said. "If diplomacy does not move in the right direction ... those measures will increase."
He added: "North Korea has previously confirmed to us its willingness to denuclearise and we are eager to see if those words prove sincere."
Kim is due to leave on Tuesday afternoon, a source involved in the planning of his visit has said.
One of the world's most reclusive leaders, Kim visited Singapore's waterfront on Monday, smiling and waving to onlookers, adding to a more affable image that has emerged since his April summit with South Korean leader Moon Jae-in.
The Swiss-educated leader, who is believed to be 34, has not left his isolated country since taking office in 2011, apart from visiting China and the South Korean side of the border Demilitarised Zone, which separates the two Koreas.
Just a few months ago, Kim was an international pariah accused of ordering the killing of his uncle, a half-brother and scores of officials suspected of disloyalty.
The summit was part of a "changed era", North Korea's state-run KCNA news agency said in its first comments on the event.
Talks would focus on "the issue of building a permanent and durable peace-keeping mechanism on the Korean peninsula, the issue of realising the denuclearisation of the Korean peninsula and other issues of mutual concern", it added.
Ahead of the summit, North Korea rejected unilateral nuclear disarmament, and KCNA's reference to denuclearisation of the peninsula has historically meant it wants the United States to remove a "nuclear umbrella" protecting South Korea and Japan.
Trump spoke to both South Korean President Moon Jae-in and Japan's Prime Minister Shinzo Abe on Monday to discuss developments ahead of the summit.
For Kim, the authoritarian leader of militarised state that has shunned contact with the outside world, the ultimate goal aside from security guarantees would be freedom and support to develop an impoverished economy.
For Trump, achieving a momentous foreign policy success would cement his place in history, and give his presidency a boost just months ahead of congressional mid-term elections.
Many experts on North Korea remain sceptical Kim will ever completely abandon nuclear weapons, believing his engagement aims to get the United States to ease crippling sanctions.
"The process could be doomed before it begins," said Kelsey Davenport of the Arms Control Association, adding that a common understanding of denuclearisation was key to success.
(Except for the headline, this story has not been edited by NDTV staff and is published from a syndicated feed.)
Get Breaking news, live coverage, and Latest News from India and around the world on NDTV.com. Catch all the Live TV action on NDTV 24x7 and NDTV India. Like us on Facebook or follow us on Twitter and Instagram for latest news and live news updates.