The 24-year-old is pushing for a move away from Old Trafford in January due to his lack of chances under Ruben Amorim so far this season.
Zirkzee has started just three of his nine Premier League games, scoring once, in their 2-1 win over Crystal Palace at the end of November.
Italian giants Roma are understood to be keen on bringing the Dutchman back to the Serie A but are yet to agree on a fee with Manchester United.
According to Corriere dello Sport, United are holding out for a permanent deal or a loan with an obligation to buy at the end of the season.
Roma have proposed a loan deal with an obligation, but only if certain conditions are met, something United are reluctant to sanction.