The 22-year-old spent the second half of the 2020-21 campaign on loan with the Gunners, scoring two goals and providing two assists in 20 appearances in all competitions.
Arsenal are pushing to bring a number 10 to the club before the end of the transfer window, and Leicester City's James Maddison and Lyon's Houssem Aouar continue to be linked with potential moves.
However, according to Italian journalist Fabrizio Romano, the North London club's main attacking target is still Odegaard, and they are hopeful of having the chance to re-sign him in the coming weeks.
Romano claims that Mikel Arteta's side want the Norwegian on a permanent deal, but Real Madrid are yet to signal whether they are prepared to let him leave.
Odegaard was a second-half substitute for Los Blancos in their pre-season friendly with AC Milan on Sunday evening, which ended in a goalless draw.