Odegaard has agreed to join the Gunners until the end of the season, with the loan deal containing no option for Arsenal to make the move permanent in the summer.
The 22-year-old has made just seven LaLiga appearances for Real Madrid this season and he is eager to gain more action in the second half of the campaign.
Odegaard had also attracted interest from Real Sociedad, where he spent last year on loan, but Arsenal have won the race for the attacking midfielder.
The Gunners have been in the market for a playmaker throughout the current transfer window and have looked at a number of targets.
Odegaard’s availability became clear last week, when Real Madrid decided he would go out on loan, and Arsenal’s technical director Edu has acted swiftly to tie up a deal.
The Norwegian is due at the club’s training ground on Monday to undergo his medical and finalise the relevant paperwork before joining Arsenal, who are thought to be paying a loan fee of around €2million (£1.8m).
Arsenal wanted to sign a creative midfielder this month to provide competition for Emile Smith Rowe, who has burst onto the scene and into the first team.
The departures of Mesut Ozil, who joined Fenerbahce on Sunday, Sead Kolasinac and Sokratis Papastathopoulos have freed up space in the squad, but Arsenal will still need to offload another non-homegrown player once Odegaard has signed.
Brighton goalkeeper Mat Ryan, who joined on loan last Friday, has taken up one non-homegrown spot in the Arsenal squad and another must be found to add Odegaard.
Goalkeeper Alex Runarsson is one player who could go on loan, while Shkodran Mustafi - who has six months left on his current deal - is attracting interest from Lazio.