The 22-year-old spent the second half of last season on loan at the Gunners, making 20 appearances in the process.
Real are interested on listening to offers for Odegaard, who has failed to stake a claim to a first team place since joining at the tender age of 16.
Since returning to his parent club, Odegaard has been blasted for his performance in Real's shock 2-1 pre-season loss to Scottish champions Rangers.
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