Sanchez struggled to make an impact at United after joining from Arsenal in January 2018 and fell out of favour under Ole Gunnar Solskjaer.
He only scored three Premier League goals for the club and has been allowed to depart despite United s lack of attacking options.
Solskjaer will only have Marcus Rashford and teenager Mason Greenwood available to face Southampton at the weekend as Anthony Martial is injured.
Sanchez will reunite with another former United forward Romelu Lukaku at Inter, who scored on his Serie A debut for against Lecce in a 4-0 win.