The Daily Star is reporting that the north London side are being offered the opportunity to land the 25-year-old because he is deemed to be surplus to requirements at the Bernabeu.
The same story says that Arsenal have been offered the chance to sign the Dominican Republic international for a cut-price £18m this summer.
Diaz found his first-team opportunities at Real Madrid to be limited last season, and he only started three League games for the Spanish side, scoring three goals in a total of 13 appearances.
It is claimed in the same story that Arsenal are on the lookout for a new forward after allowing Danny Welbeck to leave the club this summer.
Arsenal are thought to be on the lookout for a number of new signings this summer as Unai Emery prepares for his second season in charge of the north London club.
The Gunners finished in fifth place in the Premier League table and without a trophy last season under Emery as they lost the Europa League final to Chelsea FC.
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