Odegaard, 19, was billed as Europe's next top talent when he moved to 'Los Blancos' for £3m at the age of 16, though has seen his career slow down after spending time on loan in the Eredivisie.
In 18 months at Heerenveen, Odegaard featured in 43 matches and scored four goals which has earned him admirers across Europe once again.
Real Madrid want to send the Norwegian out on loan to gain first-team experience, with the Championship likely to be his next destination.
Aston Villa and Steve Bruce are still able to bring in players on loan until August 31st as part of the EFL's transfer policy.
He has failed to secure a place in Real Madrid's starting lineup, making only one La Liga appearance under Zinedine Zidane.