Several pro-Shebab radio stations broadcast what they said were the last words of Abdusalam al-Muhajir who said that he "wanted to die as a martyr" after spending the last two years fighting alongside the rebels.
"I am an American citizen who grew up in the United States after my parents took me there at the age of two," he said in the message which was broadcast a day after the attack on an African Union base in the capital.
Sheikh Mohamed Ibrahim, a Shebab spokesman, told AFP that "two young and brave mujahideen warriors, one who came from the United States, led the holy attack that killed many enemies yesterday."
The African Union peacekeeping force (AMISOM) in Mogadishu issued a statement after the attack, saying its troops had "beaten off an attack by Al-Qaeda linked terrorists on one of their positions in the outskirts of the city.
"During the failed attack, the suicide bombers blew themselves, but the extremists were unable to take control of the AMISOM position."
The statement did not indicate the number of casualties.
In their statement issued on Saturday, the Shebab claimed to have killed 80 Ugandan soldiers in a battle that lasted over two hours.
Uganda and Burundi provide the 9,000 soldiers that currently comprise the AU mission in Somalia, which is tasked with defending the fragile interim government from the insurgents.
The Shebab, who control large swathes of territory in south and central Somalia, withdrew from their positions in Mogashishu in early August following a government offensive launched in February.
They have claimed credit for a series of bombings since, including an October 4 blast at a government complex that killed 82, the deadliest single attack ever committed in Mogadishu. (*)
Editor: Kunto Wibisono
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