The suspect, Anis Amri, lived in Italy after leaving Tunisia in 2011, a Tunisian security source told AFP on Wednesday.
According to daily La Stampa, he arrived on Italy's shores by boat and told police in 2011 he was a minor, despite being 19 at the time. Italy's laws offer protection and benefits for unaccompanied migrant minors.
The Tunisian was assigned to a foster home and school in Catania, where he got in trouble for threatening and hurting other students, before eventually attempting to set fire to the school.
The arson attempt led to his arrest in October 2011. In prison he was treated as dangerous but showed no signs of radicalization, Italian media report.
An expulsion order issued after Amri had completed his sentence was blocked by red tape in Tunisia, which did not recognize him as a citizen, and Amri was able to travel to Germany.
His ID was found in the truck which drove into a crowded market on Monday evening, killing 12 and injuring almost 50 more.
German prosecutors have now issued a Europe-wide wanted notice for the 24-year-old, offering a €100,000 reward for information leading to his arrest and warning he "could be violent and armed".
As the manhunt intensified on Thursday morning, questions were raised about how the suspect had been able to avoid arrest and deportation despite being on the radar of several security agencies.
German authorities have come under fire after it emerged that Amri should have been deported from the country in June, and opposition parties linked the attack with Chancellor Angela Merkel's open doors refugee policy even before it was known who the suspect was.
If it is confirmed that Amri was connected to the attack, there is likely to be political backlash in Italy too, where record numbers of migrants have arrived by boat this year.
One Italian, a 31-year-old woman, is feared to have died in Monday's attack, while two others are reported injured, though one has already been released from hospital.
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