The order was contained in a directive from 2011 that has only now been made public, KBS TV and Yonhap reported.
It said no newborn could be given the name, and those who already had it should amend their birth certificates.
The rule is thought to have applied to leaders Kim Jong-il and Kim Il-sung, Mr Kim's father and grandfather.
Although the directive is quoted as saying that the changes should be done voluntarily, correspondents say few in North Korea would be likely to disobey.
Kim Jong-un emerged as heir-apparent during 2011, but parents naming their children in the years before would have had little forewarning of what was to come.
Such is the secrecy surrounding the ruling family, the current leader was barely known to most North Koreans until the very last days of his father's rule at the end of 2011.
As the youngest son he would not have been considered the most likely successor even by those who did know of the family.
The BBC's Charles Scanlon says having the same name as the ruler was banned in the ancient dynasties of China and its Confucian neighbours.
In North Korea, for all its revolutionary rhetoric, the ruling family demands much of the mystical reverence once reserved for the emperors of old.
North Korea has ordered anyone who shares the name of the leader, Kim Jong-un, to change it, according to South Korean media.