The Gunners are facing a battle to keep hold of their prized asset amid interest from Premier League title rivals Manchester City.
Arsenal have failed to agree a new contract with the Chile international despite opening negotiations over an extension back in 2016.
Sanchez has less than 12 months remaining on his current deal and the 28-year-old would be available on a free transfer next summer.
The former Barcelona man scored 24 times in 37 games last season but his efforts still weren’t enough to secure a top-four finish.
Arsenal spent a club-record £52m fee on Lyon striker Alexandre Lacazette earlier this month to add some more firepower to their squad.
Speaking to ESPN FC about whether Arsenal will finish in the top four next season without Sanchez, former Tottenham goalkeeper Keller had a bleak forecast for the Gunners.
“Unless Arsenal bring in several players ,if Sanchez leaves [Arsenal will finish outside the top four],” Keller said on Monday.
“If you look at the current squad with Sanchez gone, it will be difficult to finish in the top four.”
Arsenal finished in fifth place in the Premier League table last season to miss out on a place in the Champions League this coming season.
The Gunners did win the FA Cup thanks to a 2-1 victory over Premier League champions Chelsea in the final.
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