The 28-year-old looked set to be reunited with Pep Guardiola, whom he played under at Barcelona, only for his £60m (€67.6m, $81m) move to collapse on deadline day after Arsenal failed to secure a replacement.
Sanchez has turned down an offer to extend his deal at the Emirates and the Gunners are resigned to losing him either in January or for free next summer, with the striker allowed to speak with foreign clubs in the New Year.
However, Arsenal now appear eager to avoid the latter scenario and on Thursday (12 October), Arsene Wenger hinted both Sanchez and Mesut Ozil could leave in January, as there are no discussions planned with the pair over their future.
"When you are in our situation you envisage any solution. It is possible [they could go in January]," said Wenger, when asked if talks between Arsenal and the duo were in the pipeline, according to Sky Sports.
Should Arsenal be willing to part ways with Sanchez halfway through the season, however, they would have accept a bid worth a fraction of what they could have received in the summer. According to the Mirror, City are ready to test Arsenal's resolve with a £20m bid in January, but could also opt to sign him for free in the summer, when Sanchez could get a more lucrative deal.
Should he move to the Etihad in January, the Chile international would not be cup-tied as Arsenal are not playing in the Champions League this season, which would be an added bonus for Guardiola. City, however, might have to fend off competition from Paris Saint-Germain, as rumours linking Sanchez with a move to the French capital refuse to go away.
In September, it emerged Sanchez's representatives met PSG executives during the summer when it became apparent Sanchez wanted to leave the Emirates, but not even the Parisians could afford to meet his £400,000-a-week demands.
However, the Chile international was reportedly told PSG were ready to match City's £275,000-a-week offer on top of a hefty sign-on bonus.
Last week, French newspaper Le Parisien reported PSG had already set the wheels in motion to secure the 28-year-old's signature this summer. According to the publication, PSG would be willing to sell Edinson Cavani in the summer to accommodate Sanchez, which would also allow them to recoup part of the fee they would have to pay Monaco to make Kylian Mbappe's move to Paris permanent.
PSG have already spent a world record £198m on Neymar and have committed to buying Mbappe for £166m from Monaco next summer, after signing him on a season-long loan, meaning they would need to sell some of their players to comply with Financial FairPlay regulations.
Meanwhile, Wenger believes Sanchez will remain focus on doing his best for Arsenal until he leaves the club.
"I have no doubt that he wants to perform," he said. "He never goes on the football pitch to lose a football game, never. Sanchez is a winner type. You have always to set targets when you're a football player. When one is gone, you focus on something else."
The Chilean is unlikely to start when Arsenal travel to Watford on Saturday, after playing in his country's 3-0 defeat to Brazil on Tuesday (10 October) night which spelled the end of Chile's hopes of qualifying for the 2018 World Cup.