Spanish daily Marca says Real will let Odegaard leave on a one-year loan deal as coach Zinedine Zidane has not included him in his first-team plans for the coming season.
The 17-year-old Norwegian made his debut for the Spanish giants at the end of the 2014/15 season but did not make a single appearance for the club under Rafael Benitez or Zidane last campaign.
The Sun and Marca say Liverpool are Odegaard's most likely destination this summer, even though his father Hans-Erik is thought to prefer a switch to the Bundesliga.
The teenager visited Liverpool's Melwood training complex before he opted to move to Real from Stormsgodset in January 2015.
Bayer Leverkusen and Hamburg are among the German clubs reported to be interested in the prodigy.
Bernabeu clear out
In July, Spanish daily AS reported that there was conflict between Zidane and Madrid president Florentino Perez over the latter's insistence on Odegaard being included in the club's pre-season tour of North America.
Zidane was reported to not want Odegaard in his pre-season squad as he had no plans to play him in the first team in the coming season, but Perez wanted him on the tour to help Real find a club to loan him to this summer.
With the 11-time European champions facing a potential transfer embargo from Fifa, they could be forced to make decisions on the future of several of their young players before the transfer window shuts.
Marco Asensio is thought to be the only youth-team product that will be part of the first-team this season, with Enzo Zidane, Marcos Llorente and Mariano all likely to leave the Bernabeu on loan, according to Marca.
Madrid are gearing up for their Uefa Super Cup encounter against fellow Spanish club Sevilla in Trondheim on 9 August.