Left-footed Odegaard, who has played nine times for Madrid this season, mainly plays as an attacking midfielder but can also operate on either flank.
He will provide direct competition for 20-year-old Emile Smith Rowe, who has impressed in the No10 role in recent weeks, with Arsenal conscious they cannot lean too heavily on such a young player in a key role.
Despite Odegaard’s lack of action at the Bernabeu, Los Blancos boss Zinedine Zidane has said his chance would come if he stayed with Madrid.
“Let him keep working,” said the manager. “He is a beloved player and he will surely succeed here.
“He has not played much lately but the idea is to keep working hard – he will have minutes.
“He will provide insurance. This is a specific moment. In a race things happen to the players, but he is not discouraged.”