According to ESPN, the Norwegian international remains their preferred target but Lyon’s Houssem Aouar is also on the Gunners’ radar.
Mikel Arteta is known to be in the market for a playmaker to boost his team’s attacking threat after a disappointing first full season in charge.
Arsenal finished down in eighth place in the Premier League, failing to qualify for European competition for the first time since 1995.
Although defensively sound, they scored just 55 goals and were a shadow of some of the free-flowing sides constructed by Arsene Wenger.
Odegaard had a positive impact after joining on loan in January, adding much-needed spark to the Arsenal midfield.
Despite registering just two goals and two assists in 20 appearances across all competitions, he was a lively presence throughout his few months at the club.
It was believed that he would stay at Real Madrid this season, but they now appear open to selling the 22-year-old for £40million.
If that valuation proves too steep, then long-term target Aouar would come under consideration once more.
The French international came through the ranks at Lyon to become an influential figure at the club, combining composure and vision with excellent technical ability.
But the effects of the coronavirus pandemic, coupled with a failure to qualify for the Champions League, have put financial pressure on the French side.
As a result, Aouar is reportedly available for around £25millon, a fee that would largely be covered by Joe Willock’s impending departure to Newcastle United.
The Arsenal academy graduate has agreed personal terms with the Magpies and looks set to complete his move to Tyneside shortly.
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