The 28-year-old has not played for Arsenal since their derby win against Tottenham on September 1 and suffered the injury while away with Spain during the international break.
Arsenal boss Arsene Wenger had earlier predicted that Cazorla would not feature until after the October international break, but the playmaker has recovered better than expected.
And playing in Arsenal's home Champions League fixture against Rafael Benitez's side on October 1 is his target.
Cazorla told Sphera Sports: "I hope to be back shortly. Every day I'm feeling better and I'm working on all ways to reduce my downtime.
"It all depends on how I train and strengthen my ankle.
"As the days go by I am doing more things and if all goes well I will try to return for the game against Napoli, but that's not certain."
Arsenal sit joint top of the Premier League table with Tottenham, with both north London clubs having posted four wins from their five matches.
