He's also repeatedly expressed his support for a meeting between Trump and Kim - even offering to facilitate it or serve as an ambassador of sorts.
Pyongyang has taken some steps to show sincerity, returning USA detainees and blowing up its nuclear test site.
The two leaders are staying in separate hotels, not far from each other.
The two leaders meet at 9am on Tuesday at the Capella on Sentosa island, a refurbished British Army artillery mess that is now one of Singapore's most expensive luxury hotels.
The art of the unconventional deal?. I don't think this will be a one-time summit: There will no doubt be more negotiations and summits, but this is a great starting point.
"The officers have done a good job under very pressed circumstances", he said, pointing to the short notice given to prepare for the meeting. There was no need to.
"The minimum would be a relationship - you'd start at least a dialogue", Mr Trump added, when asked about outcomes.
"Sorry, we can not let our friends, or enemies, take advantage of us on trade anymore", Trump wrote in a tweet.More news: NBA Finals 2018 Game 4
The role would be as a "guarantor", Chinese analysts say, not just of progress on the denuclearisation Washington is seeking, but also to ensure what Kim wants most: the safety of his regime. "They both get what they want".
"We continue to call on North Korea to completely, verifiably, and irreversibly dismantle all of its weapons of mass destruction (WMD) and ballistic missiles as well as its related programs and facilities", it said. One dispatch by the Korean Central News Agency said North Korea and the USA would exchange "wide-ranging and profound views" on establishing new relations, building a "permanent and durable peace-keeping mechanism", achieving denuclearization and "other issues of mutual concern, as required by the changed era".
The KCNA news agency heralded the summit as part of a "changed era". In return for safeguards and potential economic help, Trump demanded "complete, verifiable and irreversible" destruction of North Korea's nukes.
Asked by Stephanopolous whether Trump deserves credit for brokering the first meeting in recent history between a sitting USA president and a North Korean leader, Menendez said he is anxious Trump "thinks this is a mano-a-mano engagement in which he can achieve the success we want".
Meanwhile, President Trump arrived at Paya Lebar Airbase in Singapore in the evening and was also received there by the country's foreign minister Vivian Balakrishnan.
"I am on a mission of peace and we're going to be carrying the hearts of millions of people from all over the world", he said.
Mr Kim arrived on a plane loaned from China along with his outspoken Foreign Minister, Ri Yong-ho, Defence Minister No Kwang-chol and his sister Kim Yo-jong.
His convoy included more than 20 vehicles, including an ambulance.
Arriving at the Prime Minister's office after 1900 local time, Kim shook hands with Lee and his staff then sat down to begin talks in front of the cameras before the press was ushered out a few minutes later. The two men are expected to discuss North Korea's nuclear and missile capabilities, which the country has rapidly advanced during the past year under Kim's watch.