Goal revealed earlier this week that the Reds would consider offers for the 32-year-old after seeing him slip down the pecking order.
Cagliari have now seen off interest from Germany to land the experienced centre-back, with a medical passed and terms agreed on a two-year contract prior to the closure of the Serie A transfer window.
Jurgen Klopp has personally sanctioned the sale – having made it clear that he will have the final say on any exits at Anfield - with Klavan in the final year of his contract and keen to play regular first-team football.
He turns 33 in October and Klopp is keen to be fair to the player, who has been a consummate professional during his time at the club.
With Joel Matip fit again after a thigh problem, and Dejan Lovren expected back at the end of the month, Klopp feels his options at centre-back are sufficient, especially with Nathaniel Clyne's return enabling Joe Gomez to move inside from right-back.
Gomez started for Liverpool against West Ham last weekend, and is seen as a centre-half, long term. That has been a key factor in the decision to allow Klavan to leave.
Klavan joined the Reds for £4m ($5m) from Augsburg in 2016 and went on to make 53 appearances, scoring twice.
His sale is likely to the be the first of a number in the next fortnight, with Simon Mignolet, Lazar Markovic, Pedro Chirivella, Marko Grujic and Divock Origi all facing uncertain futures.
Exits for all, either on loan or permanently, could be sanctioned.