The 23-year-old was named as Hassan Abdi Dhuhulow, who the BBC said is suspected of helping to plan and carry out the attack on the upmarket Westgate mall.
Dhuhulow was born in Somalia, but he and his family moved to Norway as refugees in 1999, according to relatives who spoke to the BBC from the Norwegian town of Larvik, some 135 kilometres southwest of the capital Oslo.
However, other relatives denied it was Dhuhulow who appeared in security camera footage of the attack.
"None of the men in the video is the 23-year-old," a relative told Norwegian broadcaster NRK.
The relative's name or links to the suspect were not disclosed, but they reportedly live in Larvik.
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The BBC quoted one of Dhuhulow's former neighbours Morten
Henriksen, who described the young man.
"He was pretty extreme, didn't like life in Norway... got into trouble, fights, his father was worried," Henriksen told the BBC, speaking of Dhuhulow as a teenager.
Last week Norway's PST intelligence agency said it had launched a probe after it obtained information about the possible involvement of a Norwegian of Somali origin in both planning and carrying out the attack.
Norwegian investigators have been sent to Nairobi to work with their Kenyan counterparts, Norway's PST said.
Neither the intelligence services nor the family responded to AFP phone calls on Friday morning.
Witnesses in the mall described how the fighters stormed the complex around midday on September 21 when it was crowded with shoppers, firing from the hip and hurling grenades at shoppers and staff.
The gunmen coldly executed scores of people, with witnesses recounting how in some cases they called out to those wounded, then finished them off at close range.
The siege was declared over four days later.
Kenyan police have named four of the attackers as Abu Baraal Al Sudani, Khatab Ali Khane and one man known simply as Umayr - reportedly all Somalis, plus a Kenyan of Somali origin, Omar Nabhan. However, the names are noms de guerre.
