It's the first ever meeting between a sitting USA president and North Korean leader.
"It was not easy to get here.There were obstacles but we overcame them to be here", Kim replied.
The US president has also said that if he thinks things are going badly, he will walk out of the meeting, but if things go well, Mr Kim could receive an invitation to the White House.
In this photo released by the Ministry of Communications and Information of Singapore, North Korean leader Kim Jong Un, right, arrives at the Changi International Airport, Sunday, June 10, 2018, in Singapore ahead of a summit with US President Donald Trump.
The U.S. president will then address the media before flying out late Tuesday Singapore time. "But I really believe that we have the potential to do something incredible for the world".
The meeting carries not only potentially dire diplomatic risks, but high political stakes as well for a president who has framed the meeting in legacy-defining terms. But he did not address human rights issues when he met with Kim's top aide, Kim Yong-cho.
What is unclear whether the summit will lead to any type of substantive outline of an agreement in which North Korea will halt its program and agree to denuclearize, and what kind of verification procedure it would accept.
It is an outcome that seemed unimaginable less than a year ago - when Mr Trump and Mr Kim exchanged streams of fiery insults - and North Korea conducted several ballistic missile tests in defiance of the worldwide community.
For the first time since the division of the Korean Peninsula, the leaders of the two countries held a summit in June 2000.
Trump and Kim are scheduled to talk to each other alone for about 45 minutes with only their translators in the room before their advisor join them.
Trump landed in Singapore a little after 8:30 p.m. local time for the highly-anticipated meeting with the North Korean leader, which is set to take place on Tuesday.
White House press secretary Sarah Huckabee Sanders said Sung Kim, the United States ambassador to the Philippines who has taken the lead on policy negotiations with the North, will hold a "working group" with a North Korean delegation.
But when pressed by reporters, the secretary of state would not say whether or not that could include reduction of the number of or removal of USA troops in South Korea.
The two are due to meet on Tuesday on the resort island of Sentosa for their high-stakes summit, the first-ever talks between a USA and North Korean leader and the culmination of a head-spinning diplomatic dance.