Reports say the man opened fire on police who asked him for ID during a routine patrol in the Sesto San Giovanni area in the early hours of Friday.
German authorities say fingerprints they provided have confirmed the dead man is Amri. They are trying to find out if he had accomplices.
The BBC reports that Monday’s attack on a Berlin Christmas market left 12 dead and 49 injured.
Meanwhile, the Islamic State (IS) group has released a video purporting to show Amri pledging allegiance to its leader, Abu-Bakr al-Baghdadi.
He does not make any reference to the Berlin attack, which IS claimed soon afterwards. It is not clear when or where the video was filmed.
Shortly before releasing the video, IS acknowledged Amri’s death in Milan.
When Italian police stopped the suspect, who was on foot, at 03:00 a.m. local time, he “immediately drew out a gun” and shot at the two policemen, Italian Interior Minister, Marco Minniti, said.
One of the officer, Cristian Movio, was injured in the shoulder but his injuries are not life-threatening.
His junior colleague, Luca Scata, who had been in the police for just nine months, was the one who fired the shot which killed Amri.