The youngster spent last season on loan with Real Sociedad where he netted an impressive seven goals for the Basque club.
Whilst it is claimed that both Madrid and Sociedad had an arrangement in place for the loan deal to last two seasons, it is now being claimed that Los Blancos boss Zinedine Zidane is keen to hand Odegaard a chance in the first team at the Santiago Bernabeu.
Zidane is expected to make several changes to his squad this summer.
Despite winning the La Liga title, Madrid crashed out of the Champions League to Manchester City last week and the French boss is now keen to move a number of players on.
Spanish daily Marca claim that a decision on Odegaard's future is now set to be made over the coming days.
Who could be leaving Madrid this summer?
Perhaps the biggest name to be subject to a potential summer sale is Gareth Bale. The Welshman was not selected for the Champions League tie against Man City last week and question marks remain over his future.
Colombia international James Rodriguez is another player reportedly set for a summer exit with the likes of Arsenal said to be keen on making a move.
Luka Modric is another name linked with a potential exit with Marca claiming he could be sold as Zidane looks to freshen up his squad ahead of next season.
Dani Ceballos also waiting for future decision at Madrid
Ceballos who spent last season on loan with Arsenal claimed he wanted to make a permanent move to the Emirates stadium and recent reports in the Madrid press claim a deal worth €30m-35m could be agreed.
The former Real Betis midfielder made 24 appearances last season for the Gunners, providing two assists.
A decision over his future, much like that of Odegaard, remains hanging in the balance.