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The fee is £1.3m (€1.5m) plus bonuses and Young will fly to Milan on Friday ahead of a medical, according to Sky Sports News.
Antonio Conte’s side are close to agreeing a swap deal with Roma which would see left-back Leonardo Spinazzola moving to the San Siro while Matteo Politano goes the other way but that does not appear to have impacted on a deal for Young.
The versatile 34-year-old has been keen to discuss the opportunity of a move to Italy with Inter.
United offered Young a new contract earlier this month but the former England international rejected a one-year contract extension.
Young has made 18 appearances this season, but has only started 10 league games under Ole Gunnar Solskjaer.
He joined United from Aston Villa in June 2011 and made 261 appearances for the club, scoring 19 goals.