Allegri, 57, spent four years in charge at San Siro between 2010 and 2014, winning the Serie A title in 2011.
Milan sacked Sergio Conceicao on Thursday after less than six months in the job.
The Portuguese coach helped Milan win the Italian Super Cup but the club ultimately failed to qualify for any European competition after a disappointing eighth-placed finish.
The seven-time European champions were eliminated by Feyenoord in the Champions League playoffs in February then got beat by Bologna in the Italian Cup final this month.
Meanwhile, city rivals Inter Milan will play Paris Saint-Germain in the Champions League final on Saturday.
Allegri is most well known for his two spells in charge at Juventus, during which he won five league titles and reached two Champions League finals.
