The Serie A club is keen on a deal to take the 28-year-old back to Italy in the summer, though Sunderland wants to keep the midfielder to aid the fight against relegation.
The 28-year-old has struggled to fully adapt to life in English football and would be keen on a return to his homeland despite having only made the switch to the Black Cats from Juventus last summer.
As exclusively revealed back in October, Walter Mazzarri's Inter had been interested in a deal to take the Italy international to San Siro, but Giaccherini's wages of around 49,000 euros-a-week proved a stumbling block.
Fiorentina has now emerged as the favorite to land Giaccherini although a January deal, to add to the loan signings of Antonio Rosati, Alessandro Matri and Manchester United midfielder Anderson, looks highly unlikely.
Gus Poyet is reluctant to do business during the winter window and part with a player whom he believes can make a significant contribution in the club's battle to avoid relegation from the Premier League.
Giaccherini has thus far appeared in 17 league games for the bottom-of-the-table Black Cats, although a summer move to Vincenzo Montella's Fiorentina would be an attractive option for the player, with the Italian club on course to finish the season in the Champions League spots.
Reports had linked the Azzurri star with a return to Juventus, but, speaking in December, Giaccherini's agent denied that the midfielder was looking to leave and claimed that there had been no contact with the Serie A champion.