The Dutch forward has signed a contract until June 2029, with the option to extend for a further year.
Zirkzee, the 2023/24 Serie A Under-23 Player of the Year, was Bologna FC’s top goalscorer last season, helping his team qualify for the UEFA Champions League.
The 23-year-old Netherlands international earlier won six trophies during his time with Bayern Munich.
Upon signing, Zirkzee said: "Having had discussions with the manager and the leaders of the club I know how exciting the future is going to be here and I cannot wait to play my part in achieving success for Manchester United.