“Alexis Sanchez has joined the Nerazzurri on a free transfer,” the Serie A side said in a statement.
The 31-year-old, who had been on loan at Inter this season, signed a contract until June 30, 2023, the club added.
Sanchez scored four goals for Antonio Conte’s Serie A runners-up behind Juventus.
He was sidelined for nearly three months after undergoing surgery in October on tendon damage picked up while playing for Chile in a friendly against Colombia.
Sanchez joined United as part of the deal that saw Henrikh Mkhitaryan sign for Arsenal in January 2018, but failed to make an impact at the Premier League club.
He scored five goals in 45 appearances for United before following Belgian forward Romelu Lukaku to Inter in August 2019.
“Everyone at Manchester United wishes Alexis Sanchez all the best in his future career as he joins Inter Milan on a permanent transfer,” the Premier League club said.
United had been looking to offload Sanchez, who earns around £390,000 pounds ($704,500) a week according to British media, with Inter agreeing to pay less than half his wages when they signed him on loan.
United had signed the Chile international in January 2018 after he netted 80 goals in three and a half years at Arsenal, but he failed to reach those heights at Old Trafford 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, scoring only four Serie A goals this season, 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 who could cost the Premier League side around £100 million.