The Holland international has already agreed personal terms with United after being courted by their manager Erik ten Hag.
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CBS Sports' Ben Jacobs is reporting: "Manchester United have told Bologna they are prepared to pay or match Joshua Zirkzee's €40m release clause.
"Zirkzee focused on Euro 2024 causing a delay in any decision, but his agent Kia Joorabchian is in direct talks with Manchester United. Understand the commissions spoken of are around €10m.
"Milan have refused to pay this high despite getting buy-in from Zirkzee pre-window. Zirkzee is now currently a top striker priority for #MUFC."