The three are between the ages of 18 to 27, and were all being held on suspicion of belonging to an organization linked to terrorist crimes.
Amri's nephew reportedly confessed to communicating with his uncle via the secure messaging app Telegram.
The country's interior ministry said that Amri had sent money to his nephew so he could join him in Europe.
He was arrested alongside two other men in the north African country on Friday.
Amri appeared to have packed hastily and was carrying just over 1,000 euros with him.
He said police and security forces were studying the information and deciding whether to make any arrests.
The Islamic State did not specifically name Amri when it said Tuesday that a "soldier" committed the deadly attack. And across Europe, authorities are trying to learn more about Anis Amri, a Tunisian and ex-convict that Germany had meant to deport.
"It is very important for us to determine whether there was a network of accomplices.in the preparation or the execution of the attack, or the flight of the suspect", federal prosecutor Peter Frank said Friday.
Amri, a 24-year-old, had been living in Germany as an asylum seeker.
Minitti added that an officer was injured and evacuated to a hospital. A second officer, Luca Scata, was unharmed.
Amri was killed in Milan on Friday after he apparently opened fire on police who asked for his identification papers.
So why, Italy wonders, did Europe's most wanted man end up here? The Tunisian had previously lived in Italy.
The suspect shot officer's partner in the shoulder before trying to escape, but was thwarted by the rookie officer who shot and killed him.
Security was also stepped up in central Milan and other Italian cities, particularly near major churches where faithful were attending Christmas services.
German security agencies began monitoring Amri in March, suspecting that he was planning break-ins to raise cash for automatic weapons to carry out an attack.
Ms Marine Le Pen, leader of France's far-right National Front, published a blog post that dubbed the Schengen area a "security disaster", adding that France had been "reduced to learning after the fact that an armed and unsafe jihadist was probably wandering on its soil".
De Iesu said that besides the gun, Amri had been carrying a small pocket knife.
In March, he was put on a German security services list of risky people, which now includes 549 individuals, the officials said.
German Chancellor Angela Merkel has pledged a "comprehensive" analysis of how the known militant was able to slip through the net. After completing his sentence, Amri could not be deported because Tunisia did not recognize him as a citizen, so he moved to Germany, Italian media said. It was in Italy that he was radicalised.He entered Germany posing as a Syrian refugee.