Alalshikh, a self-proclaimed Manchester United fan, subsequently called for the club's majority owners, the Glazer family, to also depart.
Ruben Amorim's dismissal on Monday morning followed a turbulent 14-month tenure, reportedly triggered by a public fallout with the club's hierarchy.
The Glazer family, American owners of Manchester United for 20 years, have long faced criticism from fans over the club's significant debt.
Alalshikh, chairman of Saudi Arabia’s General Entertainment Authority, has previously commented on the club's ownership, once sparking debate about a potential new investment deal.