The Norwegian returned to Spain following a six-month loan spell with Arsenal, where he made 20 appearances and scored two goals.
Odegaard place for a first team opportunity at Madrid is unclear and craving for more playing time, the former Heerenveen player has joined the Gunners.
It is believed that the 22-year-old could play against Chelsea in Arsenal's second game of the season on Sunday.
Real Madrid brought Odegaard to Spain from Stromsgodest in 2015, for a fee reported to be in the region of €3 million.
In developing his talent, Odegaard has spent loans with Heerenveen, Vitesse and Real Sociedad with his recent one being with Arsenal.
Odegaard made just 11 appearances in his stint with Real Madrid.
The arrival of Odegaard is expected to make Arsenal midfield a better one as the side look to build a strong team to compete for the title this time around.
After losing to newcomers Brentford in the season opener, the London-based club expect to make their first win of the season over Thomas Tuchel's Chelsea.
Odegaard had one of his best season during his spell with LaLiga club Real Sociedad where he scored 4 goals with six assists in thirty-one league games in the 2019-20 season, helping The White and Blues finished 6th in the said season.
