Zirkzee, 23, made the move to Old Trafford from Bologna over the summer, and he netted on his debut for Man United against Fulham in the Premier League, with his goal securing all three points.
The Netherlands international has since gone 16 matches without scoring, though, and there has been a host of speculation surrounding his future in recent weeks due to his struggles since arriving in Manchester.
According to Tuttosport, Man United are giving serious consideration to allowing Zirkzee to leave on loan in January, with the club believing that a short period away from Old Trafford would be beneficial.
The English club's new manager Amorim has allegedly already signed off on the departure, with the Portuguese having serious reservations when it comes to the striker.
Zirkzee 'could leave Man United on loan in January'
Juventus are said to be the club showing the strongest interest, with the Old Lady believed to have been keen to sign him from Bologna during the summer transfer window.
Zirkzee scored 14 goals and registered nine assists in 58 appearances for Bologna ahead of his move to English football, but it has been a struggle him in the Premier League this term.
Former Man United interim head coach Ruud van Nistelrooy recently said that the club would need a proven goalscorer if they were to become a force in the Premier League and Champions League again.
However, Van Nistelrooy, who has now left following Amorim's arrival, recently insisted that Zirkzee will become "a very good player" for the Red Devils once he has settled into his new surroundings.
Will Man United sign a striker in January?
Man United are believed to be keen to bring a new forward to the club during the winter transfer window, with their leading target in the final third of the field thought to be Sporting Lisbon's Viktor Gyokeres.
A mid-season move for Gyokeres has already been ruled out, but the Red Devils are said to be looking at other options on the market, with a loan deal potentially being agreed.
Paris Saint-Germain's Randal Kolo Muani is thought to be on Man United's radar, with the Frenchman, who has only scored twice in 12 appearances at club level this season, unsettled in Paris.
Chelsea's Christopher Nkunku is also believed to be under consideration; the 26-year-old has scored 10 goals in 17 appearances for the Blues this season but has only managed one goal in 10 Premier League outings, with the attacker not currently a part of Enzo Maresca's starting side in England's top flight.