The German boss acquired a host of fresh talent last summer with most of his acquisitions brought in to bolster Liverpool’s attacking options.
But Klopp is now thought to be keen to strengthen his defensive-midfield area and Inter Milan’s Kondogbia is in his sights, Transfermarketweb claim.
Kondogbia, who has five France caps to his name, impressed during his debut season in Serie A last term following his £26m move from Monaco.
But the 23-year-old’s relationship with manager Frank de Boer has soured this term after the Dutchman hauled him off after just 28 minutes against Bologna last month and publicly criticised his performance.
The Frenchman has managed just four appearances after falling down the pecking order with Ever Banega, Joao Mario and Gary Medel ahead of him.
And now Klopp is said to be willing to give him a chance to play in England and has asked Anfield chiefs to make a deal possible in January.
Liverpool are still in need of defensive improvements having conceded 10 goals this season – the most of any team in the top half of the table.
Kondogbia is under contract with Inter until 2020, but Liverpool will reportedly try and end his time in Italy early by making a move for him at the turn of the year.