After breaking through at Stromsgodset at the age of 15 Odegaard was linked with Bayern Munich, Manchester United and Liverpool before joining Real in 2015.
"Real Madrid and Vitesse Arnhem have agreed the loan of Martin Odegaard for next season, until June 30, 2019," Real said in a statement.
Odegaard's performances for Real's B side were criticised and he was reportedly unpopular with his team mates for demanding to train with the senior squad. Former Real coach Carlo Ancelotti described his signing as a "public relations exercise".
During the past season and a half Odegaard was on loan at Heerenveen in the Dutch top flight, scoring three goals in 38 league appearances.
Odegaard became the youngest Norway international when he made his debut against the UAE in a friendly at 15 years and 253 days in 2014. He has been capped 12 times.
(Reporting by Rik Sharma, editing by Ed Osmond)
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