Daily newspaper Bild reported Anis A was known to police as a possibly dangerous individual, and part of a large Islamist network.
The Islamic State group claimed responsibility for the Berlin truck attack that killed 12 and almost 50 wounded.
The militant group's Amaq news agency said "A soldier of the Islamic State carried out the Berlin operation in response to appeals to target citizens of coalition countries."
The agency did not name the attacker.
Interior minister Thomas de Maiziere was wary of the claim and said that several possible leads were being followed before arriving to conclusion.
"We just heard about the supposed claim of responsibility by this so-called Islamic State that is in fact a gang of terrorists," de Maiziere told ARD broadcaster.
"There are several leads that investigators are following now," the minister added.
The claim came after German prosecutors released a Pakistani asylum seeker, who was the main suspect, citing lack of evidence.
He was arrested late Monday after he was reportedly seen jumping out of the truck and fleeing.
The prosecuters office said, "The forensic tests carried out so far did not provide evidence of the accused's presence during the crimes in the cab of the lorry".
They believe the perpetrator was still out large.
De Maiziere had earlier told broadcaster ZDF "We can't rule out that the perpetrator is on the run", and that he was confident there would be "progress" in the investigation.
He assured "nobody will rest until the perpetrator or the perpetrators are caught".
Monday's attack was similar to the one carried out in Nice in July, when a truck mowed down people celebrating Bastille Day, killing 86. The Islamic State had claimed responsibilty for the attack.
(WION with inputs from agencies)