The Gunners have so far failed to convince the 28-year-old to pen an extension at the Emirates.
However, the Chile forward revealed before leaving north London for international action his future would be sorted by his agent.
Bayern Munich and City appear frontrunners to land Sanchez this summer.
With only one year remaining on his current Arsenal deal, the two clubs are looking to swoop.
Reports at the weekend claimed ex-Barcelona boss Pep Guardiola is ready to make Sanchez City's highest-paid player on £275,000 a week.
Alexis Sanchez could leave Arsenal for Manchester City this summer
Arsenal have not had to reject an offer from the Etihad outfit
However, according to Sky Sports, Arsenal have not received a bid from their Premier League rivals.
Arsene Wenger would be reluctant to sell his star man to City as they provide direct competition for the Gunners.
But so far, Arsenal have not had to reject an offer from the Etihad outfit.
Arsenal wages: Which stars earn the most? Tue, June 6, 2017
Express Sport brings you every Arsenal player's weekly wages (according to totalsportek.com)
Play slideshow Getty Images 1 of 28
Rob Holding: £10,000
A deal for Sanchez has been mooted at around £50million, even though he has just 12 months until his contract expires.
Sky Sports add though that Sanchez remains a key target for City, while both they and Bayern hope to learn his plans once he returns from international duty.
City have the lure of Champions League football for Arsenal's main man, while he can also reunite with Pep Guardiola.