It comes as nuclear talks between the US and North Korea broke down after the failed summit between Kim and Trump in February in Vietnam.
The US is demanding that North Korea abandon its nuclear weapons entirely before international sanctions are lifted. 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.
Kim "said with satisfaction that the letter is of excellent content," Pyongyang's Korean Central News Agency reported.
"Appreciating the political judging faculty and extraordinary courage of President Trump, Kim Jong Un said that he would seriously contemplate the interesting content," the agency said, without elaborating.
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, AP reported.
Trump's letter also came days after Kim's summit with visiting Chinese President Xi Jinping, which experts say underscored China's emergence as a major player in the diplomatic push to resolve the nuclear standoff with the North.
North Korean state media said Kim and Xi discussed the political situation surrounding the Korean Peninsula and reached unspecified consensus on important issues.
Xi is expected to meet with Trump next week in Japan during the G-20 summit. Analysts say he could pass him a message from Kim about the nuclear negotiations.
Kim said during his New Year's speech said he would seek a "new way" if the United States persists with sanctions and pressure against North Korea. After the collapse of his meeting with Trump in Hanoi, Kim said Washington has until the end of the year to offer mutually acceptable terms for a deal to salvage the negotiations.
Following a provocative run in weapons tests, Kim initiated negotiations with Seoul and Washington in 2018, which led to three summits with South Korea's president and his first with Trump in Singapore on June 12 last year, when they issued a vague statement on a nuclear-free Korean Peninsula without describing how and when it would occur. The lack of substance and fruitless working-level talks set up the breakdown of Kim's second meeting with Trump, which the Americans blamed on excessive North Korean demands for sanctions relief in exchange for only a partial surrender of its nuclear capabilities.
Nuclear negotiations have been at a standstill since then, but on the eve of the anniversary of the Singapore summit, Trump told US reporters he received a "beautiful" letter from Kim, without revealing what was written. In an interview with TIME magazine last week, Trump said he also received a "birthday letter" from Kim that was delivered by hand a day before.
Trump and Kim also exchanged letters in 2018 after their first summit. White House spokeswoman Sarah Sanders said at the time that the letters addressed their commitment to work toward North Korea's "complete denuclearization."
In September 2018, Trump told a cheering crowd at a campaign rally in West Virginia that Kim "wrote me beautiful letters and they're great letters. We fell in love."
Analysts say the gesture of sending letters is part of North Korean efforts to present Kim as a legitimate international statesman who is reasonable and capable of negotiating solutions and making deals. Because of the weight of formality they provide, Kim might see personal letters as an important way to communicate with leaders of countries the North never had close ties with, they say.