Sporting director Fabio Paratici has identified the Dutch forward as the man to replace Moise Kean, who is closing in on a move to Como worth around £34m plus £4m in add-ons.
Fiorentina's proposal to United is structured as a paid loan worth roughly £4-5m, accompanied by a buy option that would convert into an obligation if certain conditions are met. Zirkzee is understood to have already given his approval for the move to Florence, and the offer currently puts the Serie A club ahead of Juventus in the race for his signature.
While Fiorentina wait on a response from Old Trafford, Paratici has moved to cover his bases.
An official £17m bid has been lodged with Everton for forward Beto, identified as the leading alternative should the Zirkzee deal fall through. The Portugal international, previously of Udinese, is valued by Everton in the region of £17-21m and has a contract running until 2027.
Manager Fabio Grosso is waiting on the outcome before giving final approval for Kean's departure, with Zirkzee having struggled for consistent game time at Manchester United since his move from Bologna, contributing to United's willingness to consider a temporary exit.
Fiorentina are also working the flanks, with an improved offer put in for Torino winger Alieu Njie after initial contact earlier in the window. The bid is now understood to be worth around £13-14m including bonuses, with Torino holding out for closer to £17m for the 21-year-old Sweden international, who has made 18 appearances for the club over the past season.
Njie, who has asked Torino to sell him, has an agreement in principle over personal terms with Fiorentina that would run until 2031, with the move seen in Florence as a long-term investment in a still-developing wide player rather than simply a numbers replacement following Ernest Poku's exit to Besiktas.
With three separate negotiations live simultaneously - Zirkzee, Beto and Njie - the closing days of the window represent a significant reshaping of Grosso's squad as Fiorentina look to strengthen their attacking options before Tuesday's deadline.