Odegaard enjoyed an impressive six month loan with Arsenal last season, though there is growing pessimism of closing a deal with parent club Real Madrid for the Norway international. New coach Carlo Ancelotti hasn't ruled out working with Odegaard in Madrid next season.
Instead, Arsenal are ready to turn to Fekir as an alternative.
Betis will be reluctant sellers, but Arsenal intend to test their resolve with an offer.
Fekir, while with Lyon, was all set to sign for Liverpool in 2018 only for the deal to collapse over a failed medical - a claim long denied by the player's camp.
The France international, 27, has a deal with Real Betis to 2023. Last season, the midfielder played 38 games, scoring five goals and producing six assists.
- updated June 14