This is according to Barcelona-based newspaper Sport, who claim the 22-year-old is planning to leave the Bernabeu in the New Year.
Jese is frustrated with his lack of minutes this season and feels Rafael Benitez, who replaced Carlo Ancelotti in the summer, has not been fair with him.
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The report adds that Arsenal could sign the former Spain Under-21 international on an initial loan deal, with an option to buy included.
Arsene Wenger would be free to decide whether or not to sign Jese on a permanent contract next summer, based on how the striker settles in and performs at the Emirates Stadium.
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Jese’s friend and former teammate, Enrique Castano, revealed last month that the Real Madrid forward was becoming increasingly agitated with his lack of opportunities under Benitez - and it now seems that the talented striker is prepared to leave the club, where he has been since 2007, in order to play regular first-team football.
Arsenal don't really need Jese...
However, there are no guarantees that would happen at Arsenal, where Wenger already has plenty of attacking options at his disposal.
It could be argued that the Gunners - who already have the likes of Theo Walcott, Olivier Giroud, Mesut Ozil, Alexis Sanchez, Aaron Ramsey and Alex Oxlade-Chamberlain in their ranks - don’t need the Spaniard.
However, if he was content to work hard for his opportunities in the first team, there’s no doubt Jese would be a useful option for the north London outfit.
Arsenal fans: would Jese Rodriguez be a decent signing for the Gunners? Have your say by leaving a comment in the box below!