The Gunners have already rejected a £50m bid from City for the player.
But the London club will only sign-off on the deal if they can recruit another forward in the remaining hours of this summer’s transfer window.
The Gunners believe they have done good business in selling Alex Oxlade-Chamberlain for a fee understood to be £35m while in the previous year of his contract, but want to keep hold of Sanchez as they do not have enough time to sign a suitable replacement.
Arsenal were strongly linked with a move for Sterling when he left Liverpool.
Liverpool are also interested in the France worldwide, however, and sources have told Goal that the Reds are confident of securing his signature. Julian Draxler or Thomas Lemar?
One player who is leaving the Emirates is striker Lucas Perez, signed last summer for £17.1m, who is returning to former club Deportivo La Coruna on loan after scoring seven goals in 21 games for the Gunners.
Gunners boss Arsene Wenger has consistently maintained the player would not be sold but that has not stopped speculation.
Having concluded the Confederations Cup, where he revealed he had made up his mind about his future, he informed Arsenal he wanted to move to the Etihad Stadium and would not sign a new deal with the Londoners.