In recent months, it has been claimed that both the Gunners and Red Devils will be in the market for a new striker during the summer transfer window.
Although Arsenal have recently scored a glut of goals, Mikel Arteta is allegedly keen to bring in a fresh face to compete with at least one of Gabriel Jesus and Eddie Nketiah.
From United's perspective, Anthony Martial is expected to depart on a free transfer, leaving Rasmus Hojlund and the versatile Marcus Rashford as the senior options down the middle of the attack.
Whoever they each attempt to sign remains to be seen, each club having been credited as admirers of several high-profile names yet it being unclear whether their respective FFP situations will prevent them from spending so freely on one player.
Before this campaign, Zirkzee had rarely been linked with a top-end Premier League, a consequence of his development stalling after limited success when contracted to Bayern Munich.
However, a loan stint at Anderlecht during 2021-22 helped kick-start his career and he has since become a key figure at Bologna, one of the surprise packages in Europe.
According to Gazzetta dello Sport, the Italian outfit will demand €80m (£68.4m) if any clubs come calling for the Dutchman.
The 22-year-old has contributed 10 goals and three assists from his 26 appearances in Serie A this campaign, Bologna remarkably sitting in fourth position in the table.
Securing any kind of European football will be regarded as a huge success, yet Bologna are five points clear of sixth spot, with fifth potentially being good enough for Champions League qualification.
While Bologna would ideally want to demand as lucrative a fee as possible, former club Bayern do possess a buy-back clause of €40m (£34.2m) should they wish to use it.
That appears highly unlikely, as it stands, with Harry Kane the standout first choice in attack and Mathys Tel viewed as his understudy.
Zirkzee possesses a contract at Stadio Renato Dall'Ara until the end of 2025-26.