The Spanish outlet report that Madrid were reluctant to part ways with the 22-year-old last month, despite the Reds’ willingness to meet their £23m valuation.
Jese has only started seven games in all competitions this term, but made a significant impact off the bench as he scored in Real Madrid’s 2-0 win at Roma in the Champions League last Wednesday.
With Los Blancos’ squad loaded with superstars, particularly in attacking areas, the Spaniard’s first-team prospects may not improve.
Klopp was quoted in the Liverpool Echo last week as saying the Reds have “not too many wingers”, and Jese’s ability to operate from wide as well as up front seems to have caught the German’s eye.
The Reds have Daniel Sturridge and Divock Origi back fit after each of them had suffered injury setbacks, but adding another quality forward to their arsenal is in Liverpool’s best interest.