The report called Klopp a "big admirer" of the Spanish forward, who is a player capable of adding pace and a scoring touch to Liverpool's mercurial attack.
Still just 22, Jese is an enticing talent. He can stretch a defence with pace, boasts excellent close control and has a real eye for goal. The Spaniard recently demonstrated those qualities in Los Blancos' UEFA Champions League win over Serie A side AS Roma by scoring a fantastic solo effort.
B/R's Sam Tighe even thinks Jese's recent performances are pushing James Rodriguez out of the Madrid starting lineup:
What will likely appeal to Klopp is the player's versatility. Jese can operate wide or through the middle. In fact, realising his awesome potential may eventually rest on a full-time move to centre-forward.
Is Jese worth the risk, despite his history of injuries?
It's a position where Klopp needs reinforcements. Daniel Sturridge is a talisman, but one who can't avoid injury, while Divock Origi is an intriguing prospect but still raw.
Jese would fit in the type of fluid attacking trio Klopp is trying to build with the Reds. If there's one reason to doubt this move it's Jese's own history of injury woes. He's struggled for regular playing time since an anterior cruciate ligament injury suffered in 2014, per FourFourTwo's David Cartlidge.
Understandably, Liverpool won't want to rely on another brittle striker next season.
Liverpool Hoping for Lucrative Mario Balotelli Sale
There should certainly be an opening for a new striker in Klopp's squad next season, with Mario Balotelli's future seemingly destined to be away from Anfield.
The Reds are hoping a club in China will offer a hefty fee for the troubled Italy international, according to Chris Beesley of the Liverpool Echo:
Liverpool are hoping to get a big fee for Mario Balotelli as China’s top clubs eye the Italian striker armed with their new found wealth.
The 25-year-old striker is currently back on loan at former club Milan for the remainder of the season but he is not part of Jurgen Klopp’s plans and has seemingly kicked his last ball for the Reds.
Balotelli has the physical talent to make it at any top club. His pace, brawn and shooting power make him a nightmare for defences on those rare occasions he is motivated enough to dominate.
Will Liverpool get a big fee for Balotelli?
But that's not happened at Liverpool, where the player's notoriously erratic temperament has made him a major flop on Merseyside. Now the club should merely take anything they can get for the misfiring forward and forget about holding out for a hefty fee.
Whatever happens with Balotelli, Klopp still needs more forwards suited to his quick-breaking style of play. Provided Jese's durability is no longer in doubt, the Madrid man is an ideal fit.