The announcement comes ahead of Mr Trump’s visit to Seoul, South Korea, next week, when he is expected to discuss the possibility of another summit with North Korea after talks broke down in February.
Mr Kim reportedly said “with satisfaction that the letter is of excellent content” and praised the "extraordinary courage” of Mr Trump, without elaborating on what he wrote.
"Appreciating the political judging faculty and extraordinary courage of President Trump, Kim Jong-un said that he would seriously contemplate the interesting content," Pyongyang's Korean Central News Agency reported.
Nuclear talks stalled earlier this year after the US demanded North Korea abandon its nuclear weapons entirely before international sanctions are lifted.
North Korea has not responded to repeated request from the US and South Korea to resume talks since the failed February summit in Hanoi, Vietnam.
However, Mr Trump said last week he had received a “beautiful” letter from Mr Kim but did not reveal what had been written, insisting only that it was “a very personal, very warm, very nice letter”.
The White House declined to confirm that Mr Trump had sent a letter to Mr Kim.
South Korea’s presidential office said it sees the letters as a positive development for keeping the momentum of the talks alive.
North Korea’s leader has said he would be willing to take part in a third summit if the US came to the negotiations with “the right attitude and mutually acceptable terms."
Mr Kim has asked for moves towards denuclearisation to be matched by concessions from the US, such as the lifting of sanctions against North Korea.
The two leaders have exchanged letters previously in 2018 after their first summit and frequently spoken warmly of each other.
In September 2018, Mr Trump told supporters at a campaign rally that Mr Kim "wrote me beautiful letters and they're great letters. We fell in love."
The exchange of letters is thought to be part of North Korean efforts to present Mr Kim as a legitimate international leader who is reasonable and capable of negotiating deals with foreign leaders.
Agencies contributed to this report
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