The Netherlands international made the move to Man United from Bologna over the summer, and he found the back of the net on his debut for the club against Fulham in the opening matchday of the 2024-25 Premier League campaign.
Zirkzee has failed to register in his 16 appearances since, though, and there has been criticism of his performances.
There has been a host of speculation surrounding the Dutchman in recent weeks, with a return to Serie A, potentially as soon as January, believed to be a serious possibility for the centre-forward.
According to Calciomercato, Zirkzee's agent has been in talks with his close friend Giuntoli for some time to discuss a potential move to the Old Lady in January, although any deal is likely to be on loan.
Juventus 'lead the race' to sign Zirkzee in January
The report claims that the forward is wary of 'giving up without a fight' at Man United, as he is still said to be confident of demonstrating that he has the ability to turn out for the 20-time English champions.
However, having failed to settle in Manchester, a switch back to Italy is said to appeal greatly to the former Bayern Munich youngster.
Thiago Motta left Bologna to take charge of Juventus over the summer, meanwhile, with a reunion in Turin now on the cards, and the Old Lady are believed to be firmly at the head of the queue for his signature.
Zirkzee started Netherlands' UEFA Nations League clash with Hungary on Tuesday night, playing 76 minutes of the group-stage fixture, but he failed to make his mark, not having a single shot.
Should Man United let Zirkzee leave?
Having represented his country during the international break, Zirkzee will now head back to Manchester, with Man United preparing for Sunday's Premier League clash with Ipswich Town.
It is highly unlikely that the Dutchman will be in Ruben Amorim's first XI, though, and it is believed that the Red Devils could move for a new striker during the January transfer window.
If a forward can be secured in the winter market, then a departure for Zirkzee makes a lot of sense, as his switch to the 20-time English champions has simply not worked.
However, if Man United are unable to sign a new striker in January, they risk leaving themselves short by allowing Zirkzee to leave.
There have been suggestions that Marcus Rashford could be used through the middle by Amorim, and it will be fascinating to see how the Englishman fits into the new system that is expected to be implemented by the Portuguese.