A delegation from each country gathered at the Ritz Carlton on Monday, ignoring questions from a scrum of journalists as they arrived.
The talks came as Mr Trump and Kim both spent their first full day in the city-state after arriving within hours of each other on Sunday.
The pair are expected to initially meet one-on-one with interpreters on Tuesday morning for talks that could last up to two hours.
Mr Trump, the US president, begun Monday by escalating a row with allies over trade and tariffs with a familiar early morning tweet storm.
But he later held talks with Lee Hsien Loong, the Singaporean prime minister, who presented the US president with an early birthday cake. Mr Trump turns 72 on June 14.
Mr Trump reportedly said at the meeting: “I think things could work out very nicely. We appreciate your hospitality and professionalism and friendship.”
A visit to America’s embassy in Singapore followed, where Mr Trump was pictured giving a pep talk to staff.
The US president tweeted: “Great to be in Singapore, excitement in the air!” He later tweeted a video showing him chatting to Mr Lee and signing what appeared to be a guest book.
Kim’s schedule was less clear. The North Korean leader arrived on Sunday, along with his now famous jogging bodyguards, but was not seen on Monday morning.
The talks at the Ritz Carlton were attended by Sung Kim, US ambassador to the Philippines, and Choe Son Hui, the North Korean vice-foreign minister.
Mike Pompeo, the US secretary of state who has met Kim twice ahead of the summit, said the negotiations – which lasted two hours – had been “substantive and detailed”.
While the details of the discussions are unknown, both sides have reportedly focused in recent days on narrowing the gap between their starting positions on denuclearisation.
America wants Kim to give up his nuclear weapons in full before any lifting of sanctions, while the North Koreans favour a phased process where economic benefits would come in over time.
On Monday, Mr Pompeo reiterated that America wanted the “complete, verifiable, irreversible denuclearisation of the Korean Peninsula”, the stance of numerous previous US administrations.
He added: “The President is well-prepared for tomorrow’s engagement with Chairman Kim. The US position remains clear and unchanged.”
The group of advisers who joined Mr Trump for his meeting with the Singaporean prime minister included John Bolton, his national security adviser.
Mr Bolton’s presence in Singapore is especially interesting since he was the target of severe criticism from a senior North Korean official last month, who criticised his “repugnance”.
Mr Bolton, who only recently joined the administration, had suggested adopting a “Libya model” for the talks – an idea that infuriated Kim’s team, given the Libyan leader Muammar Gaddafi was eventually toppled and killed.
Other US officials attending the meeting included John Kelly, the White House chief of staff, Stephen Miller, a senior adviser, as well as Mr Pompeo himself.
The Capella hotel on Sentosa, an island a quarter of a mile off the coast of Singapore, is where the summit will take place.
The Telegraph viewed the perimeter of the venue on Monday. Entrances were guarded by officials and signs read: “Police check, comply with police orders.”
One car attempting to drive up to the hotel was stopped and the driver ushered out before it was thoroughly searched by officials blocking the entrance.
Only a handful of the more than 2,000 international journalists covering the summit will be allowed into the hotel for the summit on Tuesday.
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