Klopp has been following Dembele's progress since he was first breaking through at Rennes and knows Barca are willing to do business this summer.
TMW says negotiations have kicked off and Klopp has learned Liverpool can take Dembele on-loan with the option to buy him outright in June 2019.
The stumbling block is the price Barca are demanding to sell. The Catalans are seeking to recoup the €105m plus €40m in bonuses they shelled out for Dembele a year ago to Borussia Dortmund. Liverpool are pushing for a discount given Dembele's problems this season.
For his part, Dembele is open to working with Klopp. The France attacker doesn't feel appreciated by Barca coach Ernesto Valverde and would prefer to work with a manager who wants to coach him.