The 22-year-old Norway Captain will join the Gunners on a 5 year contract after a bid of about £34Million was accepted by Madrid.
Odegaard, who was on loan away from Madrid to Arsenal last season, had 20 appearances for the club, scoring twice and assisting twice.
The 22-year-old leaves Madrid having played only 11 times for Los Blancos as he could not find his way into the heart of Zinedine Zidane and now Carlo Ancelotti has opted to sell him.
Arsenal is hoping the Norwegian can replicate his performance at Real Sociedad where he scored seven goals and provided 9 assists.
