Bought as a replacement for Xabi Alonso, Aquilani has endured a torrid time at Anfield since signing for £20 million in 2009 but is now way down the pecking order.
He spent last season on loan at Juventus and his performances in Italy have caught the attention of Fiorentina boss Siniša Mihajlović.
The 27-year-old has also impressed on Liverpool’s Far East tour but Kenny Dalglish has already strengthened his midfield this summer and considers the Italian surplus to requirements.
His agent Franco Zavaglia said he and his client were waiting on a firm offer before moving forward.
Zavaglia said: “Anything can happen. We’ll wait and see if there is anything concrete. Alberto has an important contract and Fiorentina are moving within certain parameters.
“I don’t know if the move is possible, but with time something could happen.”
The Viola are eager to end his Anfield nightmare and bring him back to his homeland.
The Reds will only let him leave if a permanent deal can be agreed after his loan spell in Turin didn’t materialise into the permanent transfer most expected.
Aquilani is picking up £70,000-a-week and Liverpool want that off the wage bill as soon as possible as they look to continue their own summer recruitment drive.