Speaking to reporters as he left the White House for a fundraiser in the Hamptons, Trump said he could meet with Kim again to discuss denuclearization, The Hill reported.
According to the White House pool report, Trump said a "three-page letter" arrived "just yesterday."
Trump also said he would "love to show it to everyone," but declined to give it to the press.
Trump's remarks come at a time when the Pentagon says North Korea's weapons program remains a major concern, and a U.S. congressional report states diplomacy has eased tensions but has done little for denuclearization.
"North Korea...appears to be advancing its military capabilities...has conducted multiple short-range ballistic missile tests, which violate United Nations prohibitions," the report said.
Kim's tests of ballistic missiles in July and August are a response to ongoing joint military exercises in South Korea, according to Pyongyang.
Trump has previously said North Korea has the potential to become an "economic powerhouse" if it relinquishes its nuclear weapons and cooperates more fully with the United States.
Kim is focused on developing North Korea's economy, but he may have a different path in mind.
The Rodong Sinmun reported Friday the Korean Workers' Party is seeking to "strengthen" national independence through a self-developed information economy, a reference to tech-driven development.
The Rodong also said the modernization of the North Korean economy is not only for the purpose of improving production, but to "realize the historical feat of human liberation."
"The process of modernization is never a path of certainty," the paper said. "If we surrender the fate of the country to foreign powers, we become their slaves."
Kim most recently met with Trump at the border village of Panmunjom in June.