Sanchez was so desperate to find a way out of Manchester that he was even willing to have his wages cut down at some point to allow him leave Old Trafford.
It is understood United have agreed a 10-month loan deal which does not include an option to buy, with the Premier League club retaining control of the 30-year-old who still has nearly three years left on his contract.
On Thursday, Sanchez completed a season-long loan move to Inter Milan, with United agreeing to pay 40% of his weekly wages.
Inter's loan signing of the player does not include an option for the Nerazzurri to sign Sanchez permanently next summer.
The Premier League side posted on Twitter: "We can confirm that Alexis Sanchez joins Inter Milan on loan until 30 June 2020".
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Sanchez's departure on loan to Inter Milan means 17-year-old Mason Greenwood will be given plenty of game time and Solskjaer - who revealed Chris Smalling is also on his way to Italy to sign for Roma - is more than happy to put his trust in the teenager.
The manager will face questions about the 30-year-old at Friday's press conference previewing the trip to Southampton.
Sanchez found himself warming the bench at Old Trafford after Jose Mourinho lost confidence in the Chilean's once-envied capabilities, and nothing changed with Ole Gunnar Solskjaer's entry.
It's understood that Roma have held discussions over long-serving defender Chris Smalling, while Italy global Matteo Darmian is keen on a belated return home.