During the talks with South Korean President Moon Jae-in, Trump demanded that Pyongyang started the process of denuclearization and noted that he would guarantee Kim's safety.
Commenting on the summit planned for June 12, the US president said "if it doesn't happen now maybe it will happen at another time."
Earlier in the day, US Vice President Michael Pence announced that Washington would not make any concessions unless North Korea initiates "verifiable and irreversible" denuclearization.
"Plans continue to go forward for a summit. We remain open to it, we remain hopeful. But let me be very clear: nothing has changed about the policy of the United States of America. There have been no concessions offered and none given," Pence told Fox News Radio.
The vice president went on explaining that Washington wants Pyongyang to start "verifiable and irreversible" denuclearization.
However, Pyongyang stressed earlier that it was not interested in unilateral demands from the US on denuclearization and would negotiate with Trump only if he is seeking to improve bilateral relations.
The White House, however, is still hopeful that the Singapore summit between the US president and North Korean leader would go ahead as planned on June 12. Earlier in May, US President Donald Trump noted that Pyongyang had not notified Washington about the cancellation of the forthcoming meeting.