Mr Kim spoke of his “willingness” for a third summit and extended an invitation for the US president to visit the North Korean capital, the Joongang Ilbo newspaper reported.
Mr Trump said on 9 August that he had received a “very beautiful letter” from Mr Kim.
But US officials have not said anything about a second letter sent in the third week of August.
Mr Trump and Mr 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.
Their first two meetings were formal summits, the second of which, in Vietnam in February, broke down after they failed to narrow a gap between US demands for North Korean denuclearisation and a North Korean demand for relief from sanctions.
They met for a third time on 30 June in the demilitarised zone between the two Koreas and agreed to restart working-level talks but that has not happened.
Since the June meeting, North Korea has several times tested short-range projectiles.
The White House, the US State Department and the North Korean mission to the United Nations did not immediately respond to requests for comment on the report.
An unidentified director-general for US affairs at North Korea’s foreign ministry said on Monday he hoped a “good meeting” with working-level US officials would take place “in a few weeks”.
But whether a meeting would lead to a “crisis or chance” was up to the United States, the official said, calling for a more flexible approach.
“The discussion of denuclearisation may be possible when threats and hurdles endangering our system security and obstructing our development are clearly removed beyond all doubt,” the official said in a statement carried by North Korea’s official KCNA news agency.
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North Korea’s vice foreign minister, Choe Son Hui, said last week Pyongyang was willing to have “comprehensive discussions” late this month.
Mr Trump subsequently said he would be willing to meet Mr Kim at some point this year.
South Korea’s foreign minister, Kang Kyung-wha, asked about the newspaper report, said there were “detailed explanations about such a letter” but declined to elaborate.
Ms Kang said it could be “too much to expect” that Mr Trump and Mr Kim would meet before any working-level talks.