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.
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.
The IS-linked Amaq news agency said "a soldier of the Islamic State" carried out the Berlin carnage "in response to appeals to target citizens of coalition countries".
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.
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