The attacking midfielder flew into England on Sunday evening and will travel to London Colney on Monday morning to undergo his medical and put the finishing touches on his loan switch.
Odegaard will spend the remainder of the season at Arsenal, with the Gunners agreeing to pay his wages, which are believed to be around £38,000 ($52,000) a week.
They have also paid Madrid a loan fee for the 22-year-old and there is no option to turn the move permanent included in the deal.
The move will delight Mikel Arteta, who is believed to have played a significant role in convincing the Norway international to opt against a return to Real Sociedad in favor of a spell in north London.
Odegaard’s arrival will ease the pressure on the likes of Emile Smith Rowe and Bukayo Saka, while increasing the pressure on underperforming stars such as Willian and Nicolas Pepe.
The young attacker was determined to leave Madrid this month in pursuit of regular football between now and the end of the season, having made just nine appearances for the Spanish champions in all competitions this season.
Arsenal have been searching for a creative midfielder since the start of the campaign, with Mesut Ozil having been left out of the Premier League squad by Arteta.
Martin Ødegaard is now ready to sign his contract as new Arsenal player - after the two clubs completed paperworks yesterday. Loan until the end of the season, then #AFC and Real Madrid will discuss about him and Ceballos. But as of today it’s a simple loan. ⚪️🔴 #transfers— Fabrizio Romano (@FabrizioRomano) January 25, 2021
The German has now completed a move to Fenerbahce and, in his absence this season, Arsenal have struggled for creativity, although Smith Rowe has transformed Arteta’s attack since being brought into the side on Boxing Day.
The 20-year-old has set up three goals in six games since the 3-1 win against Chelsea on December 26 and found the net himself in the FA Cup win against Newcastle.
There are some who believe Odegaard’s imminent arrival might put the brakes on Smith Rowe’s impressive development, but Arteta insists every decision Arsenal make is well thought out.
“We have a very clear idea of how we want our players to develop and what they need to do that,” he said, when asked what the capture of Odegaard could mean for Smith Rowe.
“All the decisions we make are just to reinforce their development and try to give them the best possible chance.”
As well as playing at number 10, Odegaard can also operate from the wide right position, so he could provide cover for Saka, who has been performing so well on the right side of attack in recent weeks.
That could see opportunities further limited for Willian and Pepe, whose performances were once again called into question following Saturday’s FA Cup fourth-round defeat at Southampton.