Trump was flanked in the larger meeting by chief of staff John Kelly, Secretary of State Mike Pompeo and national security adviser John Bolton. "We will be tremendously successful", Trump said before their private session.
Kim's sister gained notoriety during the Winter Olympics in Pyeongchang, South Korea in February, where she met with South Korean President Moon Jae-in.
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.
It is the first time a sitting U.S. president has ever met a North Korean leader.
Kim is due to leave on Tuesday afternoon, a source has said.
There was no word on how Trump was spending the night before the summit.
"I think that very quickly I'll know that something good is going to happen", Trump said in Canada where we met leaders of other G7 economies.
Tuesday's meeting, convened at a luxury hotel on the island of Sentosa, comes just three months after Trump accepted North Korea's invitation for talks on the spot.
It also has two presidential suites, so Trump and Kim won't have to duke it out over who gets the big room.
North Korea has previously said it was willing to discuss denuclearization, but experts have expressed concern over the varying definitions of the term held by Washington and Pyongyang. The North Korean leader was educated at a boarding school in Switzerland, where he studied English as well as taking instruction in German, although his grades were reportedly less than stellar, according to Vanity Fair. This contrasted with Kim, who had comparatively less to lose, having already scored a major win through Trump's agreement to meet him.
But enough about that - check out Trump's power move in the initial handshake.
The insult came after North Korea's foreign minister Ri Yong Ho said the country may test a powerful nuclear weapon over the Pacific Ocean - a retaliation to Trump's earlier threats of military force.
A hand from the US President on Mr Kim's arm provided some warmth to an exchange being so closely scrutinised around the world. That could generate misunderstandings and a breakdown that risks armed conflict over North Korea's nuclear weapons.
The leaders and delegates then sat down to begin bilateral talks.
George Lopez, a professor at the Kroc Institute for International Peace Studies at the University of Notre Dame, said, "Each leader will proclaim to their national audiences the summit was a success", but they may see the details differently.
"Trump is a gambler and he's gambling on being able to rein in North Korea like a parent does to naughty child", Leong said.