The 30-year-old Estonian centre-back, who has won a staggering 108 caps at senior international level, has emerged as a surprise target for Jurgen Klopp, whose pre-season plans have been rocked by injury issues.
Defenders Joe Gomez and Mamadou Sakho are both doubts for the start of the 2016/17 campaign with Achilles issues, and with Martin Skrtel and Kolo Toure having been allowed to leave Anfield over the course of the past few weeks, the German is short of cover
In fact, Klopp has only new signing Joel Matip and Dejan Lovren to choose from, and with the former having limped out of their weekend win at Wigan, at least one addition is needed.
Reports claim that talks with Augsburg are on-going, but that a deal worth around £4.2m, plus add-ons, is expected before the end of the week.
Klavan, who stands at 6ft. 1”, starred for his current side as they met the Reds in Europa League last 32 action last season.
A move for the vastly experienced defender backs up claims made recently that the club would look towards the Bundesliga for a seasoned centre-back.