The forward is in the final year of his deal at the Emirates Stadium, and Gunners boss Arsene Wenger has indicated he would be prepared to let the Chile international leave for free next summer rather than cash in now, per Goal:
While there would be nothing stopping him from joining City next year, allowing him to reinforce a forward line that already boasts Sergio Aguero, Gabriel Jesus and Bernardo Silva, among others, is almost tantamount to handing the Sky Blues the Premier League title this season.
Sanchez enjoyed an exceptional campaign last year, per Squawka Football:
While Arsenal would likely receive a healthy sum by parting with him in this transfer window, keeping him would be a significant boost to their ambitions of not only returning to the UEFA Champions League but also mounting a title challenge.
Players of his calibre make winning the UEFA Europa League a highly achievable goal too, should Wenger take the competition seriously.
City aren't desperate for more firepower, but it wouldn't be surprising if they did return to test Arsenal's resolve. However, it appears the Gunners are set to hold firm.
Meanwhile, Wenger is hoping Alex Oxlade-Chamberlain will also stay put.
He is also nearing the expiration of his contract, but according to The Sun's Geoff Sweet, the Gunners manager said: "I rate him highly, and he's one of the players who is on his way up. In the last year, he has made huge progress, and I want him to stay here for a long time."
Per the Daily Mirror's John Cross, Wenger added:
Oxlade-Chamberlain has been a regular fixture at Arsenal for six years, so it's easy to forget he's still young, having only turned 24 on Tuesday.
He turned in his best campaign for some time last year, contributing six goals and 11 assists in all competitions, and he's a dynamic and versatile presence in the team.
Goal's Nizaar Kinsella reported the midfielder is hoping for wages of £150,000 per week, a figure Arsenal are reluctant to meet.
Bleacher Report's James McNicholas believes the club could be better served focusing their attentions on other players:
Oxlade-Chamberlain is worth keeping if the Gunners can agree terms with the player, but he needs to show he can maintain or improve on the contributions he made last year.
Allowing him to join a rival—Sweet made note of Liverpool, Chelsea and Manchester United's interest in the player—would be a questionable move, though, as he could be an asset if he's able to kick on.