Anis Amri’s family, speaking from his hometown of Oueslatia, said they were shaken to discover he was the prime suspect in Monday’s attack, in which 12 people were killed.
His brothers, Abdelkader and Walid, said Anis left Tunisia after the 2011 revolution and lived in Italy for three years.
He was in regular contact with his siblings on Facebook and on the phone.
In an interview with Sky News Arabia, Abdelkader said Amri did not talk to them about his life away from Tunisia.
He said: “When he left Tunisia he was a normal person. He drank alcohol and didn’t even pray. He had no religious beliefs.
“My dad, my brother and I all used to pray and he didn’t, maybe he got into this when he was in prison where he met Algerians, Egyptians and Syrians.
“The last time I spoke to him was 10 days ago, we had a brief conversation. He never used to answer with much but he did mention that he would be coming home soon.
“We expected more from him, we wanted him to develop himself and depend on himself and then help our family.
“He went into prison with a mentality and when he came out he had a total different mentality.”
Speaking to Associated Press, Abdelkader said: “If he’s guilty, he deserves every condemnation. We reject terrorism and terrorists – we have no dealings with terrorists.
“I ask him to turn himself in to the police. If it is proved that he is involved, we disassociate ourselves from it.”
Walid added he wanted to blame him “for what he has done”.
A neighbour of the family said he left Tunisia at the age of 16 or 17 after he was accused of stealing. He was sentenced in his absence to five years in Tunisia, before fleeing to Italy.
Italian media has said Anis Amri served time in prison for setting fire to a school.
He arrived in Germany in July 2015, but his application for asylum was rejected in June this year.
His deportation, however, got caught up in red tape with Tunisia, which had denied he was a citizen for several months.
Amri’s father told Tunisia’s Radio Mosaique that his son left Tunisia seven years ago as an illegal immigrant.