In a series of tweets, Mr Trump said: "In a letter to me sent by Kim Jong Un, he stated, very nicely, that he would like to meet and start negotiations as soon as the joint U.S./South Korea joint exercise are over.
"It was a long letter, much of it complaining about the ridiculous and expensive exercises.
"It was also a small apology for testing the short range missiles, and that this testing would stop when the exercises end.
"I look forward to seeing Kim Jong Un in the not too distant future! A nuclear free North Korea will lead to one of the most successful countries in the world!"
Mr Trump announced he had received a "beautiful" three-page letter just hours before it was announced that North Korea had launched yet more short-range missiles.
The stand-off on the Korean peninsula is centring on military drills taking place in the region, in which the US and South Korea are taking part - agitating Pyongyang.
Mr Kim seemed to apologise for the missile tests, whilst also adding that they would continue until the military exercises had finished.
The president appears to hint that he is also dissatisfied with the military exercises, calling them "ridiculous" and "expensive".
Mr Trump has frequently expressed his admiration for Mr Kim and repeated comments he made last week at a rally in Cincinnati that the leader has a "great and beautiful vision" for North Korea.
If the pair were to meet following the exercises, as indicated in the letter, it would be their fourth meeting since Mr Trump became president.
They last met on the North/South Korea border in June, where the two leaders historically shook hands and Mr Trump became the first serving US president to cross into North Korea.
Before that, the US walked away from nuclear talks with North Korea after an agreement between the two could not be reached.