The Chile international has spent 2019-20 on loan with Inter and has impressed post-lockdown, scoring and assisting a combined 10 goals in 12 matches.
Sporting director Giuseppe Marotta has been looking to strike a deal with United for a number of weeks and, according to Corriere della Sera, terms have now been finalised.
Inter will supposedly hand over £13.5m for Sanchez, with an announcement expected to be made after Wednesday's Europa League last-16 tie with Getafe.
The transfer will bring an end to Sanchez's six-year stint in English football, having failed to follow up his successful spell with Arsenal across a season-and-half at United, before being loaned out to Inter.
The Red Devils are believed to have been paying the majority of the 31-year-old's £50,000-a-week wages during his season at San Siro.