City looked to be hot favourites to do a deal but it is understood that two other clubs, thought to be Manchester United and Chelsea, are prepared to pay more - in terms of transfer fee, wages and other costs - to sign the player.
City went close to signing the Chile forward in August after having a bid of £55million plus £5million in add-ons accepted but a deal could not be completed as Arsenal did not find a replacement.
It was thought they could move for the 29-year-old again in January but, given their superb form and other attacking options, their preference became to wait until he comes out of contract in the summer.
An injury to Gabriel Jesus reignited the potential of a deal but City were not prepared to pay a transfer fee of more than £20million while Arsenal were thought to want £35million.
City were convinced that Sanchez did want to join them but it appears his desire to leave Arsenal this month now outweighs that.
City decided they did not want to match the figures being offered by Manchester United and Chelsea. It now seems the total package clubs are prepared to pay for Sanchez, in terms of fee, wages and potential player exchange, adds up to more than City would have paid last summer.
All the key figures at City - chairman Khaldoon Al Mubarak, chief executive Ferran Soriano, director of football Txiki Begiristain and manager Pep Guardiola - are in agreement the deal is not right for the club.