The 31 year-old has a contract which is set to expire at the end of the season unless an extension can be negotiated but it seems that the Spain international could turn it down and leave the club instead.
The likes of Chelsea, Liverpool and Juventus are also interested in Alonso with the Anfield side keen to resign him after they sold the midfielder to Real Madrid back in 2009 for a big fee of £30m.
But now Fiorentina who have already signed Bayern Munich striker Mario Gomez could now add Alonso to their ranks but if they are to sign him they will be buying the player from outside of their budget.
Fiorentina who have flourished under Vincenzo Montella have also made the acquisitions of Joaquin from Malaga, Juan Cuadrado on a permanent from Udinese and also Josep Ilicic from Palermo.