The Holland striker is frustrated with his lack of minutes so far this season under manager Ruben Amorim.
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Zirkzee has found himself overlooked by Amorim in favour of summer arrivals and fellow attackers Benjamin Sesko, Bryan Mbeumo and Matheus Cunha.
This season, he has played just 74 minutes in the Premier League without scoring or assisting. Overall, the signing from Bologna has managed just seven goals in 53 appearances for United.
The Daily Mail says Zirkzee wants to leave Old Trafford in January as he is particularly concerned about missing a place in Holland's World Cup squad.
And it's been suggested his frustration has 'significantly escalated' in recent weeks with the only option he now sees is leaving in January.