The Chile international scored 24 times in 37 games in the Premier League last season but Sanchez’s efforts weren’t enough to secure a top-four spot.
Sanchez’s future has been a source of relentless speculation over the past few months after Arsenal failed to settle on a new contract with the 28-year-old.
The South American will available to sign on a free transfer next summer with his current deal set to expire at the end of the 2017-18 season.
However, Arsenal’s title rivals Manchester City have been linked with a controversial move to sign the prolific Gunners star.
Sanchez worked with Pep Guardiola during his spell at Barcelona and the manager is rumoured to be interested in a reunion at the Etihad Stadium.
Speaking to ESPN FC about whether Sanchez will remain at the Emirates Stadium, former Arsenal forward Robson tipped Sanchez to leave this summer.
“I don’t think they will keep Alexis Sanchez and I think they will end up just outside the top four,” Robson said.
Arsenal won the FA Cup last season thanks to a 2-1 win against Premier League champions Chelsea.
The north London side haven’t won the Premier League title since 2003-04.
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