So far, the Bundesliga outfit have chosen to rebuff each of Liverpool's bids for the Guinean international, who contributed eight goals and seven assists in Germany's top flight last season.
Officials at Leipzig have been outspoken about the futures of their star players who continue to attract interest from other clubs, but it appears that will not stop the Premier League side attempting to get a deal over the line.
According to The Mirror, Liverpool are considering whether to bid over Leipzig's asking price of £70m, with Klopp seemingly prepared to spend more on the acquisition of Keita rather than entertain moves for an alternative.
After their return to Europe, Klopp is eager to sign a new central midfielder at a time when doubts remain over the future of Emre Can.
With the German having less than a year left on his contract, Juventus are said to be keen on tempting Liverpool into a sale.