In the second letter, which was passed to Trump in the third week of August, Kim spoke of his willingness to meet Trump for another summit, one source reportedly told the Joongang Ilbo newspaper.
Neither the White House, the US State Department or the North Korean mission to the United Nations responded to Reuters’ requests for comment.
Trump and Kim have met three times since June last year to discuss ways to resolve a crisis over North Korea’s missile and nuclear programmes, but substantive progress has been scant.
The pair last met at the Demilitarised Zone (DMZ) between the two Koreas in June and agreed to restart working-level negotiations that had been stalled since an unsuccessful second summit between the two leaders in Vietnam in February.
Last week’s round of short-range missiles came just hours after Vice Foreign Minister Choe Son Hui said North Korea was willing to have “comprehensive discussions” late this month.
Trump subsequently said he would be willing to meet Kim at some point later this year.
Trump said on August 9 he had received a three-page “very beautiful letter” from Kim and added he could have another meeting with him