The 22-year-old has scored 11 goals in 29 games so far this term, showcasing his potential.
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Per Tuttosport, United are set to make a bid in the summer window, but will have to stump up a lot of cash.
Bologna do not have any financial pressure to sell and would prefer to start a bidding war.
Arsenal are another Premier League side that have been linked to Zirkzee recently.