The Gunners have as yet been unable to persuade the 28-year-old Chile international - who scored 30 goals for the club in the 2016/17 season - to sign a new contract beyond his current deal which expires in a year.
Sanchez said he would leave his future in the hands of his agent as he prepares for this month's Confederations Cup with his national side, as both City and Bayern Munich hope to learn of the forward's plans once he returns from international duty.
Arsene Wenger signed a new two-year contract to remain as Arsenal manager following their FA Cup final win over Chelsea, and has consistently maintained both Sanchez and Mesut Ozil will not be sold.
Sanchez worked with Pep Guardiola at Barcelona, and the City manager admitted in April a number of clubs would be closely monitoring the player's contract situation at Arsenal.
Full-backs are still a key area to Manchester City's summer business, though, as Benjamin Mendy remains on their target list even though Monaco want a reported £40m for his services.
City are also unaware of Tottenham having significantly increased their valuation of England right-back Kyle Walker, contrary to reports that Spurs want £60m rather than £40m for the player.
Meanwhile, Jesus Navas and Nolito are attracting interest from clubs in the Premier League and around Europe, Sky Sports News HQ understands, though no developments on the pair's future are expected this week.
