Kim also said he would wait until the end of the year "for the US to make a courageous decision" on another meeting, after his most recent summit with Trump in Vietnam broke down and both sides left without agreement.
Washington has blamed the February deadlock on the North's demands for sanctions relief in return for limited nuclear disarmament, but Pyongyang said it had wanted only some of the measures eased.
In a speech to Pyongyang's rubber-stamp parliament Friday, Kim said the Hanoi meeting had made him question whether Washington is "genuinely interested" in improving its relations with Pyongyang.
"We are willing to give another try if the US offers to have a third summit with the right attitude and mutually acceptable terms," he said, according to a report by North Korean state media outlet KCNA.
Kim added that his personal relationship with Trump remained strong, adding they could "write letters to each other" whenever they wanted.
"We will wait with patience until the end of the year for the US to make a courageous decision," Kim said.
Pyongyang has been careful not to criticise Trump personally while attacking sanctions on North Korea as an attempt to "destroy modern civilisation and turn the society back in a medieval dark age".
Trump and Kim held their first landmark summit in Singapore last June, where the pair signed a vaguely-worded agreement on the "denuclearisation of the Korean peninsula".
But the failure to reach agreement at their second summit in Hanoi has raised questions over the future of the accord.
Trump said Thursday he was mulling a third summit with Kim ahead of a meeting with his South Korean counterpart Moon Jae-in at the Oval Office -- a move supported by Seoul.
Moon brokered the US-North Korea talks and has been pushing for the resumption of inter-Korean economic projects, but doing so would fall foul of international sanctions imposed on Pyongyang.
"Our government will do what we can in order to maintain the current momentum for dialogue," Moon's office said in a Saturday statement.
The day before Kim accused Seoul of acting as an "overstepping mediator" and said the South should "speak up" for Pyongyang's interests.
On Friday, KCNA reported Kim was re-elected as chairman of the State Affairs Commission, the North's most important decision-making body, to cheers and loud applause from the delegates.