The German has already secured the signing of Chelsea striker Dominic Solanke, while Roma winger Mohamed Salah is edging closer to a move to Anfield.
But Klopp still needs to strengthen in defence, in particular at left-back.
And according to the Mirror, Celtic's Tierney could be the ideal fit.
The 20-year-old enjoyed a stellar campaign under Brendan Rodgers last term as he helped the Hoops to the Scottish Premier League title and he's already been tipped for a bright future.
His impressive displays have already alerted the likes of Arsenal and Manchester United with both clubs said to be tracking him.
Celtic, though, will be keen to keep hold of their star man, who is valued at around £15m.
Tierney is under contract with the Scottish champions until 2021.
The Mirror add that should a move for Tierney prove to be unfeasible they could go after Hull's Andrew Robertson.
Source: Daily Express
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