The Mirror says there has now been direct contact between the clubs – and City have made the shock offer involving Sterling.
England star Sterling has started the season well, scoring the winner at Bournemouth and equaliser against Everton.
But this will be a major surprise for Sterling as City look to shake up their squad and pull off a major deal.
City believe a player-plus-cash deal is the only way that Arsenal may be tempted to sell Sanchez.
Arsenal may only be tempted by around £70m but City have not offered that in straight cash.