The 30-year-old has been identified as the ideal back-up centre-back by Jürgen Klopp and is due on Merseyside on Monday evening.
He will have his medical the following day with the Reds keen to complete the move before Thursday so that Klavan can travel with the first-team squad for their tour of the USA.
Liverpool's interest in the Estonia international, who has made more than 100 appearances and currently captains his national side, was only made public at the end of last week with Klopp looking to add depth to his defence.
Injuries to Mamadou Sakho and Joe Gomez, who are both in danger of missing the start of the season, combined with Tiago Ilori's involvement for Portugal at the Olympic Games had left the Reds boss short on options, with Kolo Toure released and Martin Skrtel sold to Fenerbahce.
But he views Klavan as the perfect player to provide centre-back reinforcements after he was an everpresent throughout their record-breaking Bundesliga campaign in 2014-15, finishing fifth to qualify for the Europa League.
The Reds of course met Augsburg in the last 32 of the competition last term in February, with James Milner's second-leg penalty the only goal separating the two sides after 180 minutes.
Reds hoping to wrap up quick deal for Klavan
He will become Klopp's fifth permanent signing as manager, following Loris Karius, Sadio Mane, Joel Matip and Marko Grujic - the latter two joining up with their new teammates earlier this month after sealing moves back in January.
Liverpool are also pressing ahead with a move for Newcastle United's Georginio Wijnaldum, though they will need to work on bringing the Magpies' £27 million valuation of the midfielder down.
Reports in the North East have suggested the Dutch international could even be due on Merseyside in the coming days, although sources close to Liverpool indicated on Sunday that a fee had yet to be agreed.
As such, it could be a few days yet before Wijnaldum has talks with the Reds - who are, albeit, said to be keen on wrapping up a deal as quickly as possible.
Wijnaldum wants a return to the Premier League, after Newcastle's relegation to the Championship last term in which he ended as their top goalscorer with 11, but Liverpool will likely need to pay as much as £25 million to conclude a deal.
Klopp will also hope that he can reach an agreement with Newcastle to allow Wijnaldum to join up with the rest of his squad as they travel to California this Thursday for three fixtures in North America.