Manchester United and Arsenal are both being linked to the youngster from Serie A.
Per Tuttosport, the frontman who has 11 goals in Serie A and 12 in all competitions could be available for the right price.
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United and Arsenal are both chasing a forward in the summer, albeit for different reasons.
The Red Devils want an alternative to Rasmus Hojlund, while Arsenal want a prolific scorer.
The Gunners may be the more appealing club at present, but Zirkzee may wonder if he will play more there or at United.