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Alexis moved to the Serie A club on a season-long loan in August 2019, making 29 appearances and scoring four goals.
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Since moving to Old Trafford from Arsenal in January 2018, the 31-year-old struggled to find his best form and has now been allowed to leave, relinquishing the club's iconic number seven shirt.
The Chile international has joined Antonio Conte's side permanently on a free transfer.
Sanchez leaves United having scored just five goals for the club since his swap deal from Arsenal in exchange for Henrikh Mkhitaryan two-and-a-half years ago.
Disappointing times at Man Utd
Sanchez netted 80 goals in three-and-a-half years at Arsenal, but he failed to reach those heights in Manchester under both Jose Mourinho and Ole Gunnar Solskjaer.
The former Barcelona man had said last year that he had no regrets about joining United but wished he had played more minutes, even as fans were left frustrated by his performances and lack of goals.
While he did not return to his prolific best at Inter, he did turn provider to top the club's charts with eight assists as they finished runners-up behind Juventus.
United are already working towards signing Sanchez's replacement, with the club being linked with Borussia Dortmund's promising 20-year-old forward Jadon Sancho.
Additional reporting from Reuters.
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Alexis: I wanted to return to Arsenal after my first Man Utd training session