The club face stiff competition from several clubs including Real Sociedad and Sevilla but are confident of winning the race after agreeing to Madrid's terms which included no option to buy the 22-year-old at the end of the season.
Odegaard originally joined Real Sociedad on a two-year loan last season but impressed so much that Madrid decided to cut the loan short to given him a chance at the Bernabeu.
However, the attacking midfielder has started just five matches and is seeking a move to guarantee more first-team football.
Sources have told ESPN that Odegaard missed training on Friday and has been given the go-ahead to finalise a loan deal.
Real Sociedad have not given up hope that he will decide to join them given the positive experience he had at the club but sources have told ESPN that Arsenal feel Odegaard wants to test himself in the Premier League and is favouring a switch to north London.
Talks between Real Madrid and Real Sociedad has been taking place since December but stalled over a number of issues.
La Real wanted an option to make the deal permanent but Arsenal have indicated they are happy to proceed without such an arrangement, reflecting in part the promising emergence of Emile Smith Rowe in a similar position.
Arsenal boss Mikel Arteta has faced criticism over his team's lack of creativity at times this season and the hope is Odegaard will provide a new injection of flair. The club have identified an attacking midfielder as key target this season.
They failed in a bid for Lyon midfielder Houssem Aouar last summer. Lyon are reluctant to sell in this window and with Aouar waiting to see if Real or Barcelona formalise their interest, the Gunners have decided to find a short-term solution and reassess the situation at the end of the season.
Money is also tight at Emirates Stadium as the club feels the effects of COVID-19 but they have freed up space on the wage bill by allowing Mesut Ozil, Sead Kolasinac and Sokratis Papastathopoulos to depart.