But when pressed by reporters, the secretary of state would not say whether that could include reduction of the number of or removal of US troops in South Korea.
Trump even joked that he may be wrong about Kim but will never admit it.
Secretary of State Pompeo said Monday that negotiations with the North Koreans were "moving quite rapidly", and the USA anticipated they would "come to their logical conclusion even more quickly than we had anticipated".
"The fact that our two leaders are sitting down face to face is a sign of the enormous potential to accomplish something that will immensely benefit both of our peoples and the entire world".
More officials took part in the working lunch, with the additional participants on the US side, including White House Press Secretary Sarah Sanders, National Security Council Senior Director for Asia Matt Pottinger, and senior diplomat Sung Kim, who has led substantive talks with Pyongyang into the summit.
He called Kim "very smart" and a "very worthy, very hard negotiator".
Kim said he wanted to express his gratitude to President Trump for making the meeting happen.
Trump didn't go into further detail about any agreements made but did seem very enthusiastic about the future. Assuming that nuclear war doesn't break out.
The North Korean delegation included Kim Yong Chol, a top aide to Kim Jong Un who recently met Trump at the White House.
They said the summit aimed to work towards the "promotion of peace, prosperty and security of the Korean Peninsula and of the world".
It is the first time a sitting United States president has ever met a North Korean leader. Mr. Trump "gets along so well with ruthless, brutal dictators, but you have this acrimonious, hostile relationship with the Canadian Prime Minister?"
Then they dined on beef short rib confit along with sweet and sour crispy pork. Last week, Trump told reporters that Rodman had not been invited to the summit, but praised him as a "nice guy" and great rebounder.
He added that there has been "a lot of progress".
Kim was heard telling Trump through a translator: "I think the entire world is watching this moment".
Trump responded to such commentary on Twitter, saying: "The fact that I am having a meeting is a major loss for the USA, say the haters & losers". But he added "our hostages" are back home and testing, research and launches have stopped.
O'Keefe said there may be "measured support" from some Republicans in Congress for what Mr. Trump did in Singapore, but skepticism as well.
Mr Trump said he felt "really great" about the meeting.
Employees at a foreign exchange trading company in Tokyo watch the meeting.
Trump and Kim may well end up branding tomorrow's summit a success. In a rambling interview, he claimed credit for the summit taking place and described how he had received death threats when he first met Kim.
Twitter CEO Jack Dorsey Blasted For Patronizing Chick-Fil-A
Erica Joy added: 'Please delete this or follow up with how much free advertising you're going to give GLAAD'. "You're right. One user was journalist Soledad O'Brien, a former CNN anchor, who also chose to add her two cents to the conversation.
OBikes Are Leaving Melbourne
This comes after the Environment Protection Authority announced steep new fines of $3000 per dumped bike, payable by the business. It's even once seen a bike hung up in a Melbourne alley like the scene from E.T.