The Paris Saint-Germain player has spent the last two seasons in the French capital, where he played 70 league games, scoring six goals.
Speaking ahead of his medical, the defender told Milan Channel that he wants to help the club get back to the highest level and admitted PSG team-mates Thiago Silva and Zlatan Ibrahimovic spoke highly of the club.
"I'm very happy to be here, everything is wonderful. I want to take the club back to the Champions League.
"I can't wait to get out onto the field with this shirt. Ibrahimovic and Thiago told me great things, that this club is number one in many respects."
The Rossoneri are reported to be interested in Alex's PSG team-mate Jeremy Menez, but the Brazilian insists he doesn't know if the Frenchman will join.
"I have not spoken to Menez, but I know that there is a mutual interest. I hope everything goes well."
Alex moved to Paris in January 2012 for €5 million from Chelsea, where he had won the 2009 FA Cup as well as a league and cup double in 2010.
The 31-year-old, who featured in the 2008 Champions League final while with the Blues, then went on to win Ligue 1 with PSG in both of his seasons at the club, in addition to the Coupe de la Ligue and Trophée des Champions last term.
The stopper has 17 international caps for Brazil, but has not played for his country since 2008, and was not named in their final 23-man squad for the World Cup.
Former Chelsea team-mate David Luiz, meanwhile, has previously confirmed his move to French capital, where it seems he will now replace his countryman.