Former Fiorentina coach Italiano had joined Bologna in 2024 and led them to their first Coppa Italia title in 51 years in his debut season. Bologna finished eighth in Serie A this season and reached the quarter-finals of the Europa League.
"Italiano has informed the club that he considers his time at Bologna to be over, following two seasons marked by excellent results, regardless of any future career prospects," Bologna said in a statement.
"The club would like to thank Vincenzo for the passion and dedication with which he has carried out his work over the past few years, which will remain in the club's history."
Napoli, who finished second in Serie A, confirmed the departure of manager Antonio Conte earlier this week.
Local media reports have also linked former AC Milan coach Massimiliano Allegri with a move to Napoli, along with Italiano.