The country was "ready to detonate a self-reliant A-bomb and H-bomb", KCNA quoted him as saying.
If true, the development would mark a significant advancement in North Korean nuclear capabilities.
But the claim has not been independently verified and has drawn scepticism from experts.
Mr Kim made the remarks as he inspected a historical military site in the capital Pyongyang.
The work of his grandfather Kim Il-sung had turned North Korea into a "powerful nuclear weapons state ready to detonate a self-reliant A-bomb and H-bomb to reliably defend its sovereignty and the dignity of the nation", he is quoted as saying.
October 2002: North Korea first acknowledges it has a secret nuclear weapons programme
October 2006: The first of three underground nuclear explosions is announced, at a test site called Punggye-ri
May 2009: A month after walking out of international talks on its nuclear programme, North Korea carries out its second underground nuclear test
February 2013: A third nuclear test takes place using what state media calls a "miniaturised and lighter nuclear device"
May 2015: Pyongyang claims to have tested a submarine-launched missile, which are more difficult to detect than conventional devices
While North Korea has made previous claims about its nuclear weapons capabilities this is thought to be its first reference to an H-bomb.
Such devices use fusion to create a blast far more powerful than a more basic atomic bomb.
North Korea has carried out three underground nuclear tests before, but experts cast doubt over the latest suggestions.
John Nilsson-Wright, Head of the Asia Programme at Chatham House, was sceptical, saying it fitted into a previous pattern of bold claims from the North Korean leader.
The comments were likely to be an "attention-grabbing effort to assert North Korean autonomy and his own political authority", he told the BBC.
"It's hard to regard North Korea as possessing an H-bomb," Lee Chun-geun, a research fellow at the Science and Technology Policy Institute in South Korea, told Yonhap.
But he added: "I think it seems to be developing it."
Independent observers are rarely allowed access to the secretive communist state, making verifying the authorities' claims difficult.
North Korean leader Kim Jong-un has appeared to suggest his country possesses a hydrogen bomb, in comments published on state media.