The Chile international will stay with Antonio Conte’s Serie A side until 30 June 2020.
Sanchez, 30, will link up with ex-United team-mate Romelu Lukaku, the striker having joined Inter for £74m this summer.
“On behalf of the entire Nerazzurri family, we’d like to wish him a very warm welcome and the best of luck,” Inter said.
— Inter (@Inter) August 29, 2019
He is the highest earner at Old Trafford, and it is believed United are subsidising his wage – around £400,000 a week – to allow the move to Italy.
Sanchez has had a disastrous 19 months at Old Trafford since leaving Arsenal in January 2018, scoring just five times in 45 appearances.
He scored twice last season and only once in the Premier League and, had he stayed at the club, would have predominantly played in the Europa League and Carabao Cup.
Manager Ole Gunnar Solskjaer wants to give more opportunities to teenage forward Mason Greenwood, who has made three substitute appearances so far this season.
The former Barcelona man’s form at United has been in stark contrast to that enjoyed at the Emirates Stadium, before heading north in a deal which saw Henrikh Mkhitaryan go the other way.
Sanchez scored more than 70 goals in three-and-half-years with Arsenal, and the season before joining United he was nominated for the PFA Players’ Player of the Year award after scoring 30 goals and registering 17 assists.
He has also won the Copa America twice with Chile.