The man is aged 21 or 23 and known by three different names, according to reports in the daily Allgemeine Zeitung and the Bild newspaper.
Both said asylum office papers believed to belong to the man were found in the cab of the truck.
The documents, which announced a stay of deportation, were found under the driver’s seat of the 40-tonne lorry that barrelled through the Christmas market in the heart of the German capital.
Police were reportedly searching for the suspect, who was born in the southern Tunisian city of Tataouine, in the western state of North Rhine-Westphalia.
Twelve people were killed in what German authorities have called a “terrorist attack” in Berlin late Monday, including the Polish driver of the truck.
The scenes instantly revived nightmarish memories of the July 14 truck assault in the French Riviera city of Nice, where 86 people were killed by a Tunisian Islamist.
There was no evidence to back the claim, nor was the perpetrator identified.
Tunisia is one of the biggest suppliers of jihadist fighters, with some 5,500 of its nationals believed to be involved in combat in Syria, Iraq and Libya.
© 1994-2016 Agence France-Presse
"Have no fear of perfection - you'll never reach it." ~ Salvador Dalí