Verona sacked Cioffi on Tuesday after they suffered their sixth league defeat in a 2-1 loss to Salernitana. The club are 18th in the table with five points after nine matches.
Bocchetti represented several clubs during his playing career, including Verona as well as Russian sides Rubin Kazan and Spartak Moscow.
After retiring from playing, he was part of Verona's first-team coaching staff and also managed their youth sides.