The Telegraph reports that Jurgen Klopp's hunt for a back-up centre-half is set to end with news that Klavan will join for a fee of around £3.5million.
Young defender Joe Gomez sustained an achilles injury last week and is likely to be sidelined for a little while which has prompted Klopp to return to the transfer market to reinforce his defensive options.
Estonia national team captain Klavan only needs to finalise the formalities of the deal before becoming a Liverpool player.
The 30-year-old, who played under Everton boss Ronald Koeman at AZ Alkmaar, has been with Augsburg since 2012 and has over 120 matches of Bundesliga experience while also featuring in the Europa League during his career.
Kolo Toure was released earlier this summer, Martin Skrtel has joined Fenerbahce and Mamadou Sakho is also battling an achilles problem, making the expected signing of Klavan very timely for the Reds ahead of their pre-season trip to the USA which kicks off on Thursday.