24-year-old Amri was shot dead during a routine check
The nephew of the Berlin attacker has been arrested in Tunisia, as investigators in Europe hunt for possible accomplices and seek to establish his movements.
The interior ministry in Tunis said in a statement that three men were held on suspicion of being part of a "terrorist cell...connected to the terrorist Anis Amri".
The arrests took place on Friday.
Amri, a 24-year-old Tunisian, died in a shootout with Italian police on Friday, but his death left several questions unanswered.
In Germany hundreds of investigators are set to work on the case throughout Christmas to find out if Amri was part of a larger group of jihadists or acted as a "lone wolf".
Meanwhile in Italy, investigators were trying to determine why he went to Milan and what he was doing wandering around 3.00am near a station on the outskirts of the city, where he was shot dead by a policeman on Friday after a routine check.
Italian investigators are trying to ascertain whether Amri was there to meet someone, or whether he was merely seeking shelter in the area.
Amri was believed to be the driver of the truck that sped into Christmas market in the German capital on Monday, killing 12 people and injuring scores of others.
He had filmed a video in which he pledged allegiance to Islamic State.
Chancellor Angela Merkel has expressed relief that the man, who had been on the run for four days, no longer posed a threat, but warned that "the danger of terrorism in general endures".
She pledged a "comprehensive" analysis of how the known jihadist was able to slip through the net in the first place.
"The Amri case raises questions," she said. "We will now intensively examine to what extent official procedures need to be changed."
Amri's free travels across three European countries - Germany, France and Italy - in the aftermath of the rampage are stirring calls for a tightening of the borders and a revision of the passport-free Schengen zone.
"How could Europe's most wanted terrorist leave Germany?" asked the respected Die Welt daily on its website, as German authorities face growing criticism of how they handled the probe.
Amri went to Italy in 2011 and spent over three years in prison for a series of offences.
During that time, he may have been radicalised. Newspaper Corriere della Sera said he celebrated the Charlie Hebdo attacks in Paris and threatened a fellow inmate "because he was a Christian".
Amri arrived in Germany late last year and unsuccessfully sought asylum.
He was seen as a potential threat long before the attack this week - and was even kept under covert surveillance for six months.
But authorities failed to deport him because he lacked valid identity papers and Tunisia initially denied he was a citizen.
He used at least six different names and three nationalities in his previous travels around Europe.