Odegaard spent the second half of last season on loan at Arsenal and impressed during his time in north London.
The attacking midfielder returned to Madrid this summer after his temporary spell at Emirates Stadium expired.
Odegaard had been expected to fight for his place under new Madrid boss Carlo Ancelotti, but his chances look limited.
The Norwegian is now ready to consider other options and Madrid have confirmed they are open to selling.
Arsenal have been tracking Odegaard’s situation all summer and have entered into early talks with the LaLiga giants.
The Gunners have also been keeping tabs on Leicester midfielder James Maddison, who is tempted by such a move.
Arsenal are set to bank over £20million in the coming days following Joe Willock’s move to Newcastle.