Mr Kim praised the letter as "excellent" and said he would "seriously contemplate the interesting content", the KCNA news agency said.
White House spokeswoman Sarah Huckabee Sanders said at the time that the letters addressed their commitment to work toward North Korea's "complete denuclearization".
Talks have been stalled since the collapse of a second summit between Kim and Trump in Hanoi in February after the pair failed to agree on what the North would be willing to give up in exchange for sanctions relief.
The U.S.is demanding that North Korea abandon its nuclear weapons entirely before global sanctions are lifted.
South Korea's presidential office said it sees the exchange of letters between Kim and Trump as a positive development for keeping the momentum for dialogue alive.
He also praised Trump's "extraordinary courage".
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The letters were provided Thursday to The Washington Post by the University of MS , which houses Eastland's archived papers. The former vice president is notorious for saying odd things that range from inappropriate to daft to comedic to offensive.
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However in recent months Mr Trump has spoken warmly about Mr Kim. North Korea is seeking a step-by-step approach in which moves toward denuclearization are matched by concessions from the US, notably a relaxation of the sanctions.
China's role as North Korea's vital trading partner and sole security backer gives Xi leverage in his talks with Trump.
North Korean state media has angrily denounced the USA position since that summit but has refrained from criticizing Trump directly and even referred to the positive relationship between the two leaders, a sentiment consistently echoed by Trump himself.
Trump said in an interview in May that Kim had wanted to remove only one or two of the five nuclear sites during their Hanoi summit, a deal-breaker for the US president.
"If Kim Jong Un succeeds in convincing President Trump to make a deal on past North Korean nuclear facilities, and succeeds in keeping the current nuclear missiles (he possesses) for another few years, then it would mean ... that North Korea can be accepted as a new nuclear state in this region", Thae said.