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Van Dijk, 29, could miss the entire season after he suffered an ACL injury in his right knee in Liverpool's 2-2 draw with Everton last month.
Klopp is expected to see the return of Fabinho after the international break and the club have also been linked with former Watford defender Daryl Janmaat, who is currently a free agent.
However, the Reds would most likely have to wait until next summer to sign the 22-year-old Frenchman for around £40m.
Upamecano, who signed a new deal in the summer, has a clause in his contract that allows him to leave Leipzig should his release clause be met.
Liverpool could eye up other alternatives in January should Klopp require urgent defensive reinforcements in the absence of Van Dijk and Gomez.
The Reds have been linked with Brighton's Ben White, who was subjected to two bids from the Premier League club in the summer.
According to The Mirror, Liverpool would have to offer around £30m to tempt Brighton into selling the 23-year-old in January.
Liverpool's two other reported defensive targets on their shortlist are Schalke's Ozan Kabak and Torino's Gleison Bremer.
Featured Image Credit: PA
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