The Gunners are reportedly keen on a loan deal with an option to buy the young Spanish forward.
The forward spent last season on loan at Italian side Napoli and was hoping for increased game time under Rafa Benitez this season upon returning to the Bernabeu.
He has, however, found that hard to come by, managing only 132 minutes so far in the 2015/16 campaign, and unlikely to see much more given the strong form of Karim Benzema.
That has led to increased interest from Arsenal, reports Sport, with the Gunners keen on taking the Spaniard on loan in January with an eye for completing a permanent deal should he adapt well to life in England.
According to the report, the Gunners had been keen on a deal in the summer transfer window, but instead opted to give a chance to Costa Rican Joel Campbell upon his return from Villarreal.
He has, however, once again failed to impress Arsene Wenger, who wants to add another lively forward to his side, and sees Jese as the perfect option to explore in January.