The 24-year-old has struggled for regular first-time football this season, first under Ruben Amorim and now Michael Carrick.
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Zirkzee has started just four of his 18 Premier League appearances, scoring two goals and providing a single assist.
According to TMW, United are willing to let Zirkzee leave this summer for a fee of around €35 million, with a return to Italy likely.
Juventus are said to be very interested in signing Zirkzee, who has also been linked with Roma and AC Milan in the past.