Sakho's future at Liverpool looks untenable having not made a single first-team appearance this season.
But Klopp has not included Sakho in any match day squads this season, opting to utilise Lucas as a central defender instead.
Mamadou Sakho could still leave Liverpool before the transfer window closes tomorrow
Sakho has not played a single minute for Liverpool's first team this season
And Get French Football News are reporting that Southampton have initiated discussions with Liverpool over a possible transfer for Sakho.
The report claims that Southampton are hoping to secure a loan deal with an option to buy but Liverpool want to get the player off their books permanently.
A possible stumbling block for Southampton is Sakho's £105,000-a-week wages at Anfield.
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Antonio Conte has been putting his Chelsea players through their paces at Cobham
Spartak Moscow are reportedly also in contact over a shock transfer but the report claims that Sakho wants to remain in England.
The 26-year-old broke his silence on his exile from the Liverpool squad yesterday, admitting he broke the rules in the summer.
And he went on to rule out a move to Marseille because of his allegiance to Paris Saint-Germain.