Identity documents have been found under the driver’s seat of the lorry which ploughed into a Christmas market on Monday, Der Spiegel said.
According to the report the document is in the name of a Tunisian national called Anis A, who was born in 1992 and is also known by two aliases.
A 23-year-old Pakistani asylum seeker was detained and released because of a lack of evidence in the aftermath of the attack in the German capital.
On Tuesday night the so-called Islamic State claimed responsibility for the attack which also injured 50.
Eleven people died when the articulated lorry careered through huts and stalls at the market near Kaiser Wilhelm Memorial Church, along with a Polish man found shot dead inside the lorry’s cab.
Six of the 12 victims have been identified as German.
German media said a police operation is now under way in the North Rhine-Westphalia state.
The authorities have said the ‘’modus operandi’’ of the attack had echoes of July’s atrocity in Nice in the south of France, in which 86 people died, and could have been the work of Islamic extremist groups, with the target of the attack ‘’highly symbolic’’.