The 31-year-old joined Rovers in 2009, but with only a year left on his current contract the Frenchman looks set to exit Ewood Park.
It is understood that the defender is hoping to receive a pay-off from the Championship club in return for agreeing to leave.
Once that has been agreed then the player will be able to join Livorno as a free agent.
Givet has been in Italy for talks and a medical although the Tuscany-based club released a statement confirming that a deal has not yet been reached.
"In reference to negotiations for the possible transfer of Gael Givet, AS Livorno Calcio point out that no financial agreement has been reached with the French player's club," read the statement.