Roma have been in talks with United for the unsettled Holland international, though progress has stalled over the terms.
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The Giallorossi want to setup a loan-to-buy arrangement, while United are insisting on a straight sale in January.
For his part, Zirkzee is ready to leave, despite a run of games in recent weeks.
Meanwhile, La Gazzetta dello Sport says Milan chief Igi Tare is also interested and sees Zirkzee as a replacement for Santi Gimenez.
Should Gimenez be sold, Tare would target Zirkzee as his No1 choice to replace the Mexican.