Jong-un was born to Ko Yong-Hi, Kim Jong Il's late wife.
She had another son, Kim Jong-chol, but Kim reportedly does not favour the middle son as a possible leader.
However, Japan's Yomiuri newspaper citing unnamed US intelligence sources, reported earlier on its website that Kim Jong-nam, Kim's eldest son, is expected to be appointed to a figurehead role as nominal head of state.
Kim Jong-nam, 38, had long been considered the favourite to succeed his father - until he was caught trying to enter Japan on a fake passport in 2001, reportedly telling Japanese officials he wanted to visit Tokyo Disneyland.
His mother is the late actress, Sung Hae-rim.
|The North Korean government has denied reports that Kim has been ill [AFP]
Rumours have swirled for years that Kim would nominate one of his three sons as a successor, following a dynastic tradition that began when he inherited the leadership from Kim Il-sung, his father and the country's founder.
The elder Kim died in 1994, but he remains the country's "eternal president" and the mausoleum housing his embalmed corspe is a place of pilgrimage for North Koreans.
Reports that Kim Jong-il may have suffered a stroke in mid-August have also heightened speculation about a successor.
The Yomiuri also said that Jang Song-thaek, Kim Jong-il's brother-in-law, has been assigned to look after the eldest son and is playing a central role in building a collective leadership system to back him up.
Kim Jong-il took over as leader when his father died in 1994 and rules the country with absolute authority.
However, there have been no confirmed public sightings of him for several months, raising speculation he may be seriously ill or may have already died.
A possible successor to Kim may also impact on how the country handles its relations with the West, particularly through its nuclear weapons programme.
In 2005, the North agreed to abandon its nuclear weapons programme in return for aid and other concessions.
However, the process has been stalled for months because of a standoff with Washington over how to verify Pyongyang's past nuclear activities.
North Korean state media has repeatedly denied that its leader was ever ill, and since early October has sent a steady stream of photos depicting an active and healthy Kim making visits to farms, factories and military units.
But the photos and reports are undated, and South Korean officials say they cannot confirm the visits.