Odegaard made a positive impression at the Emirates in the second half of last season
Odegaard returned to Madrid and was hoping to get a chance under Carlo Ancelotti this season, but talks have progressed on a deal to bring the Norway international back to Arsenal on a permanent basis according to the PA news agency.
While it is understood that the switch is not yet complete, the clubs have reportedly agreed a deal in the region of £30million for the attacking midfielder.
Odegaard joined Madrid as a 16-year-old in January 2015 from Stromsgodset in his homeland and later spent two seasons on loan at Heerenveen, before temporary stints with Vitesse Arnhem, Real Sociedad and Arsenal.
The Norwegian’s arrival would bolster Mikel Arteta’s options, having so far brought in Ben White, Nuno Tavares and Albert Sambi Lokonga this summer.
Arsenal have been keen to add a number 10 in recent months, with Leicester’s James Maddison and Houssem Aouar of Lyon linked.
The Gunners, who kicked off the season with a chastening 2-0 loss at Premier League new boys Brentford and are preparing to host Chelsea on Sunday, also appear to have resurrected their interest in Sheffield United and England goalkeeper Aaron Ramsdale.
Blades manager Slavisa Jokanovic says Ramsdale was “simply not in the right state of mind” to play on Wednesday night amid reports of the goalkeeper being set to join the North London club.
The 23-year-old has been linked with Arsenal since the Sheffield club’s relegation from the Premier League last season and being part of England’s Euro 2020 squad this summer.
“We felt he was not ready for the game,” Blades boss Jokanovic told Sky Sports ahead of his side’s Sky Bet Championship game at West Brom on Wednesday.
“We took the decision not to use him for the game.
“He’s simply not in the right state of the mind. We prefer not to complicate things and start with Mike (Verrips).”
Dutch goalkeeper Michael Verrips started at the Hawthorns, with Ramsdale seemingly having played his last game for United.
It has been reported that Arsenal are set to agree a £30 million deal for Ramsdale – an initial £24m, with a possible further £6m in potential add-ons – and that the former Bournemouth goalkeeper could sign in time to face Chelsea in the Premier League on Sunday