The president made the comments in a tweet on Sunday as US officials arrived in North Korea and met their counterparts at the border village of Panmunjom.
“Our United States team has arrived in North Korea to make arrangements for the Summit between Kim Jong Un and myself,” he said.
“I truly believe North Korea has brilliant potential and will be a great economic and financial Nation one day. Kim Jong Un agrees with me on this. It will happen!”
State department spokeswoman Heather Nauert said a “US delegation is in ongoing talks with North Korean officials” inside the demilitarised zone between North and South Korea that was created at the end of the Korean War.
Mr Trump withdrew from the planned summit last week, but quickly announced it could get back on track.
He said on Saturday the US was still considering the 12 June date for the summit in Singapore.
“We’re doing very well in terms of the summit with North Korea,” Mr Trump said at the White House.
Our United States team has arrived in North Korea to make arrangements for the Summit between Kim Jong Un and myself. I truly believe North Korea has brilliant potential and will be a great economic and financial Nation one day. Kim Jong Un agrees with me on this. It will happen!
— Donald J. Trump (@realDonaldTrump) May 27, 2018
“It’s moving along very nicely. So we’re looking at June 12th in Singapore. That hasn’t changed. So, we’ll see what happens.”
Mr Kim believes a summit with Mr Trump will be a landmark opportunity to end decades of confrontation, South Korea’s President Moon Jae-in said following his surprise meeting with the North Korean leader.
“He (Kim) also expressed his intention to put an end to the history of war and confrontation through the success of the North-US summit and to co-operate for peace and prosperity,” Mr Moon told reporters.
A statement from North Korea’s state news agency KCNA said Mr Kim expressed “his fixed will” on the possibility of meeting Mr Trump as previously planned.
Mr Kim and Mr Moon held talks in the demilitarised zone separating their two nations on Saturday in a scramble to salvage the summit.
Mr Trump had rattled the region on Thursday by cancelling his meeting with Mr Kim, citing “open hostility” from the Pyongyang regime.
But within 24 hours he reversed course, saying it could still go ahead after productive talks were held with North Korean officials.
During his two-hour meeting with Mr Kim, Mr Moon said he urged both Washington and Pyongyang “to remove misunderstandings through direct communication and to have sufficient dialogue in advance through working-level negotiations on the agendas to be agreed upon at the summit”.
“Chairman Kim agreed on that,” he added.
Pictures from the summit showed Mr Moon and Mr Kim shaking hands as well as embracing, and revealed Mr Kim brought along his sister, Kim Yo Jung.
She has played a major public role in talks with the South and led the delegation to the Winter Olympics in February.
Mr Trump’s decision to pull the summit is understood to have followed comments from North Korean diplomat Choe Son Hui, who referred to US Vice-President Mike Pence as a “political dummy”.
She also said it was up to the Americans whether they would “meet us at a meeting room or encounter us at nuclear-to-nuclear showdown”.
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The Trump administration is demanding that North Korea completely and irreversibly dismantle its nuclear weapons programme.
Mr Kim and Mr Trump’s initial decision to meet followed months of war threats and insults between the leaders over the programme.