Sporting confirmed in a statement that Manchester United have agreed to pay 11m euros (£9.25m) to trigger a release clause in Amorim’s contract.
Amorim’s first fixture at Manchester United is set be on 24 November, after the international break, in the Premier League against newly promoted Ipswich.
His first home game will be against Norwegian side Bodo/Glimt in the Europa League on 28 November, with a league game against Everton the following weekend.
Manchester United will announce who will be on Amorim’s coaching staff at a later date.
After losing against rivals Porto in the Portuguese Super Cup on 3 August, Amorim has guided the side he led to last season’s Primeira Liga title through a 14-game unbeaten run.
Amorim is due to be in the dugout on Friday when Sporting host Estrela in a league game (20:15 GMT).
Sporting are also at home in the Champions League on Tuesday when they play Manchester City. Amorim’s last game in charge is set to be against former club Braga in the league on 10 November, before European top-flight football pauses for Nations League games.