The 34-year-old will arrive in Italy on Friday for a medical before signing an 18-month contract, Gazzetta Dello Sport reported.
United will receive 1.5 million euros ($1.6 million) plus bonuses for Young who has spent the past eight-and-a-half seasons at Old Trafford since arriving from Aston Villa in 2011.
Young, who has played 39 times for England, has made 261 appearances for the Premier League club and scored 19 goals. He was club captain this season but only started 10 league games.
He won the 2013 Premier League title, the FA Cup in 2016 and both the League Cup and Europa League in 2017.
Young leaves the Premier League with United in fifth, 27 points behind leaders Liverpool, who they play at Anfield on Sunday.
Inter signed Romelu Lukaku from United last summer and also have Alexis Sanchez on loan from the club.
Antonio Conte’s side are second in Serie A, two points behind champions Juventus