The 22-year-old endured a challenging debut campaign wit Bologna last season, but he has since developed into one of the most prolific strikers in Serie A this term.
Indeed, Zirkzee has contributed with 11 goals and six assists in 29 appearances across all competitions this season, including 10 strikes and four assists in the top flight to help Bologna climb into the top four and challenge for Champions League qualification.
Zirkzee's performances are understood to have caught the eye of several top European clubs ahead of the summer transfer window including Man United, who are known to be on the hunt for a new centre-forward.
The Red Devils allegedly considered making a bid for the Dutchman during the January transfer window, but a move ultimately failed to materialise.
Now, Tuttosport claims that Man United had scouts in attendance at Gewiss Stadium to watch Zirkzee score a penalty in Bologna's 2-1 victory away against Atalanta BC in Serie A last Sunday.
The report adds that representatives of Premier League rivals Arsenal watched Zirkzee score the winner in a 2-1 away victory over Lazio at the Stadio Olimpico two weeks earlier.
Zirkzee is under contract at Bologna until June 2026 and the Rossoblu are believed to value their star striker in excess of €60m (£51.3m).
However, Zirkzee's former club Bayern are said to have inserted a 50% sell-on clause in his Bologna deal when they sold the striker in August 2022, while they also possess the option to buy him back for €40m (£34.2m).
Earlier this week, Bayern chief Karl-Heinz Rummenigge talked up the qualities of Zirkzee and refused to rule out the possibility of the club activating the buy-back clause for the striker, although new sporting director Max Eberl is set to make the final decision.
"I remember my last year as CEO, he played well and when Flick put him on the pitch he always scored heavy goals, not the fourth or fifth of the match," Rummenigge told La Gazzetta dello Sport.
"He made us win important matches: I'm happy for him that he's doing well, he deserves it, I've always seen him play well for us. His bad luck was having someone like [Robert] Lewandowski in front of him."
"We have just signed a new [sporting director] Max Eberl. I don't want to meddle in his work, he must demonstrate that he has the great qualities that I believe he has I don't want to influence him.
"I don't know if we will activate the clause for Zirkzee or not. It will be part of our summer strategies".
Zirkzee, who is yet to earn his first senior cap for the Netherlands, is hoping to end the 2023-24 campaign strongly in a bid to make the national team squad for Euro 2024.