The Holland international has just helped Bologna qualify for the Champions League, but could follow coach Thiago Motta out of the club this summer.
Zirkzee is interesting Juventus, PSG and also Arsenal.
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Il Corriere dello Sport says agent Kia Joorabchian's close relationship with Gunners sporting director Edu could prove decisive.
Arsenal have tabled a four-year contract to Kia and the striker, which includes an increase in his wages.
Bologna insist Zirkzee will only leave for his 40m buyout clause.