The centre-back had been expected to move on this summer, as his deal is due to expire in June 2023 and he was promised an exit if the right offer came along.
However, Inter and Juve hesitated, preferring other targets and stalled by a lack of sales, while the Corriere dello Sport have Arsenal making a proposal too.
The asking price for Milenkovic is €20m including add-ons, but the situation has changed over the last few days.
It seems increasingly likely that the Serbia international will sign a new contract with Fiorentina until June 2025.
Radio Bruno report the 24-year-old will also get a substantial pay rise at €3m per season, becoming the biggest earner at the club.
The new deal could be signed as early as Thursday, before the Serie A season kicks off this weekend.