The Holland striker passed his medical at United on Friday and met with manager Erik ten Hag and his new teammates.
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United and Bologna have agreed a fee of €40m rising to €45m in bonuses.
Zirkzee has signed a five-year deal with the option of another year. There are also suggestions he will be given the No9 shirt.
Transfer expert Fabrizio Romano reported: "Joshua Zirkzee deal now also being confirmed by his agent Kia Joorabchian’s company Sports Invest…
"…all set for official announcement as deal is done and signed."