Jurgen Klopp has told Liverpool he wants to sign Jese this summer, Sport reports.
The Spanish media outlet claim that Klopp is a big fan of the Real Madrid winger, and sees him as part of his Liverpool team for the future.
The 22-year-old has only started four La Liga games this season, and although he is highly rated by head coach Zinedine Zidane, his lack of chances is frustrating the winger.
The report believes that Jese will leave the Bernabeu this summer, with more offers expected from both England and Italy in the summer.
The report claims that Jese will be sold for roughly £23.2 million (€30m), a fee which Liverpool are willing to part with for the attacker.
Klopp has urged his board to start making contact with Madrid and Jese to speed through a move, hoping to tie him down before other clubs enter the race for him at the end of the season.
Jese has spent his entire career at Real Madrid, playing 74 first-team games. He has scored 14 goals and registered 11 assists for the club, as well as being capped five times at Under-21 level for Spain.