The Sun says Gunners boss Arsene Wenger has already started planning for a future without Sanchez, who is in the final year of his contract at the club.
Zaha, 25, is reportedly top of the Frenchman's wish list to replace the Chilean forward.
The Ivory Coast international has scored two goals in five Premier League appearances for the Eagles this season. He signed a five-year contract extension at Selhurst Park in May.
Zaha left Palace to join Manchester United in 2013 but rejoined them two years later after two loan spells at the south London club.
Arsenal may face competition for Zaha from rivals Tottenham, whose manager Mauricio Pochettino is understood to be an admirer of the winger. Manchester City were also believed to have made contact with his representatives during the summer.
Sanchez has refused to sign a contract extension at the Emirates and reports suggest City will make a fresh attempt to sign him in January.
The former Barcelona forward came close to joining Pep Guardiola's side in a £60m deal on transfer deadline day, but the move broke down as Arsenal failed to convince Monaco winger Thomas Lemar to join them as a replacement for the Chilean.
Former Gunners captain Tony Adams last week urged Wenger to sell Sanchez in January to avoid losing him on a free next summer.
"We used to have a philosophy at Arsenal: If you don't want to be at the club, then bye-bye, we shake your hand, get people that do, whatever money," he was quoted as saying by Manchester Evening News.
"I've always loved those principles, so if he doesn't want to be here, then try and get the most you can for him and get him on and get someone that does want to play for the club."