Despite having an option to trigger a 12 month extension on Klavan's contract, which has a year left to run, the Reds are open to the idea of letting the experienced defender leave in a quest for more game-time.
Klavan has been solid if not spectacular since arriving from Augsburg in 2016, however the signing of Virgil van Dijk and promotion of Joe Gomez has seen him slip further down the pecking order, now seemingly Liverpool's fifth choice centre-back.
Although a popular member around the squad and with Klopp, Klavan turns 33 in October and may wish to spend the final years of his career playing regular football as opposed to watching from the stands at Anfield.
Do Liverpool have the depth?
However, Liverpool fans may consider it a risk given the injury prone nature of their defence, Joel Matip and Dejan Lovren both currently side-lined with injuries whilst Van Dijk and Gomez have suffered lengthy lay-off's in the past.
Nat Phillips played a surprising amount in pre-season and was in line for a starting berth against West Ham if Gomez hadn't been passed fit, he'd shoot up the pecking order if Klavan were to leave.
Where he'd go would be interesting to see, although it wouldn't be hard to imagine that Augsburg would be interesting in recapturing the man who made 125 appearances for them over four years.