The Chilean is into the final 12 months of his contract at the Emirates and was heavily linked with a move away from the club during the summer.
The 28-year-old reportedly saw a deadline day move to Manchester City collapse after Arsenal failed to secure a deal for Monaco's Thomas Lemar who had been earmarked as Sanchez's replacement.
Speaking on 'The Debate', former England international Neville has questioned the decision to keep hold of the forward, saying that the Arsenal fans may have lost faith in the star.
"How will the fans react to Alexis Sanchez? He's the hero at Arsenal - he's the one that shows the grit and determination,¨ Neville explained
"They'll be showing up at The Emirates thinking 'he didn't want to play for us last week'.
"If I was an Arsenal fan, I wouldn't be supporting Sanchez tomorrow unless he showed me he was still committed to the cause."
Former Arsenal striker Charlie Nicholas agreed with Neville, saying that it may have been better had Sanchez left the club on deadline day.
"I thought they should have sold him - just get him out now...Sanchez was Arsenal's best player by a mile last season - but needs must.
"Sir Alex Ferguson got rid of Eric Cantona when the time was right as he had to. He had to find a formula. Arsenal are in a big enough mess without having a dressing room that is all over the place. If you lose your spirit and soul in the dressing room, it's the end of the line.
"He might be a warrior in some people's eyes but last season I saw him stand on the left wing for six games, and he didn't bother his backside. It was concerning for me. My concern was that Arsenal were trying to get him out and he's the difficult one to come back in - his team-mates will start asking whether he's putting it all in or is he camouflaging it."
Sanchez came off the bench for the final 15 minutes of Arsenal's 3-0 victory over Bournemouth on Saturday.