of £4.2m.
The Estonia captain, 30, would act as cover for Frenchman Mamadou Sakho, who is expected to miss the start of the season with an Achilles tendon injury.
Manager Jurgen Klopp is keen to strengthen his defensive resources after Kolo Toure was released and Martin Skrtel sold to Fenerbahce.
Klavan would be the Reds' fourth signing of the summer.
Klopp has already signed Mainz goalkeeper Loris Karius, Schalke centre-back Joel Matip and Southampton forward Sadio Mane this summer.