It was expected that Pep Guardiola's side would go for the Chilean in January and bid a cut-price £20m ($26m) fee with Sanchez's contract expiring in the summer.
However, according to the Mirror, City are still not pleased with the way the Gunners handled negotiations over Sanchez on deadline day of the summer transfer window and will instead look to come back next summer and sign him on a free transfer.
After initially standing firm on their stance that the 28-year-old would remain at the Emirates Stadium, Arsenal accepted a late £60m ($78m) offer from City only to pull out when they failed in their pursuit of a replacement in Monaco's Thomas Lemar.
The turn of events was said to have left Sanchez with a taste of bitterness according to fellow compatriot Claudio Bravo but he has remained with the club, for now, having featured in eight games this season with no signs of a contract renewal.
Things could still yet change as the same Mirror report adds that Guardiola could table an offer in January if one of his forwards suffer a long-term injury or if another club tries and pursue the former Barcelona man.
However, it is expected that they will mostly pursue him in the summer as City is said to be Sanchez's first choice due to his past relationship with Guardiola and desire to reunite with the Spanish manager.
Sanchez most recently featured in Arsenal's 2-1 win over Swansea on Saturday (28 October) and has three goals and two assists in eight games in all competitions this season.