In remarks to reporters at the White House, Trump said that the letter Kim sent him was "friendly" and included "birthday wishes", the Efe news reported.
Trump turned 73 on June 14.
After winning the 2016 election, Trump became the oldest elected US President, after Ronald Reagan, who governed from 1981-1989 and was 70 when he took office.
In a statement, White House Press Secretary Sarah Sanders said that Trump answered Kim in a letter of his own, adding that the "correspondence between the two leaders has been ongoing."
North Korea revealed that Kim had received a "personal" letter from Trump last weekend with "excellent" content, according to a brief news announcement on Monday by Pyongyang's state-run KCNA news agency.
The report did not specify the date on which Kim received Trump's missive, but it said that - after reading it - Kim expressed his satisfaction with what the US leader had written.
That letter seems to come in response to another that Kim had sent to Trump recently - which the president called "very warm" and a positive sign that contact between the pair will continue - commemorating the first anniversary of the historic summit they held in Singapore on June 12, 2018.
At that time, both leaders signed a joint declaration in which they promised to work toward the full denuclearization of the Korean Peninsula.
However, the lack of progress on that matter in the months after the summit led to a second summit, this one in Hanoi, last February.
That meeting ended early because of the lack of agreement between the two sides on how to achieve denuclearisation. Pyongyang wants a gradual process accompanied by the progressive lifting of US sanctions while Washington's stance is that it will only eliminate the sanctions when Kim ends and dismantles his nuclear program.
Since then, North Korea has toughened its rhetoric vis-a-vis the US and South Korea, with exchanges between the sides apparently at a standstill despite last year's rapprochement.
( With inputs from IANS )