Injuries to Mamadou Sakho and Joe Gomez prompted Klopp to look to the Bundesliga for reinforcements, with Estonia captain Klavan emerging as a key target.
The 30-year-old has spent the past four seasons with Augsburg, and has one year left on his current contract.
According to reporters including the Liverpool Echo‘s James Pearce, this should lead to his move to Liverpool, in a deal worth around £4.2 million.
A 6’1″, left-footed centre-back with considerable experience of top-flight football, Klavan could prove to be a shrewd acquisition for the Reds.
He is clearly a player familiar to Klopp, with the 49-year-old having encountered him during his time with Borussia Dortmund, as well as in both legs of last season’s Europa League last-32 clash.
Klavan is said to be seen as the “ideal replacement Kolo Toure,” with the former AZ defender five years Toure’s junior.
His arrival at Anfield will likely see both Tiago Ilori and Andre Wisdom to depart, with the duo reportedly surplus to requirements for 2016/17.
Both Ilori and Wisdom have featured heavily throughout pre-season so far, but Klavan’s arrival could see the latter omitted from Klopp’s squad for Liverpool’s tour of the US.
As defensive cover, Klavan seems a sensible target for Klopp, with his willingness to play a squad role key.