Insua has left the club's training camp in Switzerland to have talks with the Serie A outfit.
His imminent departure, combined with Aurelio being released after his contract expired, means manager Roy Hodgson's priority is finding a new left-back as soon as possible.
"We have agreed a fee for the sale of Emiliano Insua to Fiorentina and given them permission to speak with the player," said a club spokesman.
"Emiliano has now left our Swiss training camp and we'll make further comment when appropriate."
Insua arrived at Anfield in January 2007 from Boca Juniors and played 62 matches for the Reds.
Liverpool's friendly against Al-Hilal in Switzerland this evening has been called off after the pitch was deemed unplayable following heavy rain.
It means Hodgson will have to wait until Wednesday, when they face Swiss side Grasshoppers, for his first game since taking over from Rafael Benitez on July 1.
Hodgson thanked Insua for his service to the club.
"I am expecting him to join Fiorentina next week."