Not much detail has been revealed regarding the content of the letter, but North Korea's Central News agency has reported that Kim Jong-un "said with satisfaction that the letter is of excellent content."
"Appreciating the political judging faculty and extraordinary courage of President Trump, Kim Jong-un said that he would seriously contemplate the interesting content," KCNA said.
Nuclear talks between Washington and Pyongyang are deadlocked after a second summit between Mr Kim and Mr Trump broke up in Hanoi in February with the two unable to reach a deal on sanctions relief and what the North might give up in return.
Excellent relationship
Before the collapse of the Hanoi summit Mr Trump had said that he and Mr Kim had an excellent relationship, and had even "fallen in love."
Mr Kim recently played host to Chinese President Xi Jinping, who wrapped up a highly symbolic visit to the nuclear-armed North on Friday.
Kim Jong Un Meets Putin in Russia With U.S. Talks Faltering Source: The New York Times
Mr Kim told the Chinese president that his visit was an opportunity to demonstrate "the immutability and invincibility of the DPRK-China friendship before the world", KCNA said, using the abbreviation of North Korea's official name.
Pyongyang pulled out all the stops to welcome Xi, the first Chinese president to visit in 14 years -- a period in which North Korea has carried out five nuclear tests and launched missiles capable of reaching the entire US mainland.
Analysts say Xi intended to use the trip as a way of signalling to Trump his influence with Kim one week before the G20 summit in Japan.
