The 70-year-old, who was not on the North Korea trip, was active in Trump's feud with Canadian Prime Minister Justin Trudeau over the weekend. He told reporters he thinks he will know nearly immediately whether a deal can be made, saying: "I will know, just my touch, my feel".
"While a summit between Trump and Kim would be historic, it is unlikely to be decisive". He described it as a "get to know you plus" meeting.
Hours earlier, a jet carrying Kim landed.
"It's hospitality that we would have offered them, and as Chairman Kim said yesterday, he would have liked to have come to Singapore anyway, with or without the summit", Singapore Foreign Minister Vivian Balakrishnan told the BBC.
After the greeting at the Capella resort on Singapore's Sentosa Island, the two leaders walked to a room at the hotel and sat with American and North Korean flags behind each of their chairs.
"I think things could work out very nicely", Trump said in a Tuesday meeting with Singapore's Prime Minister Lee Hsien Loong.
After Trump's comments, Kim, speaking through a translator, said the meeting showed the United States and North Korea have overcome past obstacles.
Trump is set to meet with Lee on Monday.
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The MFA also said Lee will meet President Trump separately on Monday.
In part, these lowered expectations are a reflection of the extreme skepticism among many that the North can be persuaded to give up a nuclear programme it has stubbornly built over the decades, often in secrecy and despite intense sanctions, worldwide condemnation and widespread suffering among its people.
He accompanied the young leader at all four of his recent summits, twice with South Korean President Moon Jae-in and twice with Chinese President Xi Jinping.
Experts believe the North is on the brink of being able to target the entire United States mainland with its nuclear-armed missiles, and while there's deep skepticism that Kim will quickly give up those hard-won nukes, there's also some hope that diplomacy can replace the animosity between the USA and the North.
After a sudden and welcome turn to diplomacy following last year's threats, insults and fears of war - remember "fire and fury" and "dotard"?
The Associated Press reports Trump was originally scheduled to depart early Wednesday morning, however he will now leave Tuesday night after a day of meetings with Kim and other leaders of North Korea. But no one can credibly suggest that a durable or verifiable denuclearization of the Korean peninsula can be achieved when North Korea remains a completely closed and totalitarian state.
At 11.30am Trump and Kim will take part in a "working lunch".