According to the channel, Ikrimah al-Muhajir, 28, sought asylum in Norway in 2004 but disappeared in 2008, before a decision was taken on his application, when the US began to identify him as a potential terrorist.
"This man is an al-Shabaab bomber," Morten Storm, a former militant Islamist who personally met Ikrimah, said. "His background and origins are Somali, but he has Norwegian papers. I met him in Kenya."
According to documents from Kenya’s National Intelligence Service, seen by the UK's Daily Telegraph, Ikrimah, also known as Abdulkadir Mohamed Abdulkadir, was the leader of a terror cell which also included the British extremists Samatha Lewthwaite, who is wanted by Kenyan police, and Jermaine Grant, who is already on trial in Nairobi.
The responsible for planning attacks on a UN compound in Mogadishu and the airport in the city.
The New York Times reports that al-Muhajir, who is also known as Abdi Kadir Mohamed Abdi Kadir, was not at home and so escaped the attack on the town of Barawa, south of Mogadishu, which US troops aborted after they encountered heavy firing.
A Swedish-Somali al-Shabaab member was reportedly killed, although Swedish authorities could not confirm this on Sunday.
No US soldiers were killed in the fighting, which lasted more than an hour.
According to TV2, al-Muhajir goes under more than a dozen names, and has regularly changed his appearance to avoid detection, growing his hair long and cultivating a moustache.