The Chilean coached the Argentine goalkeeper during his time at the Spanish club, which coincided with his rise to become one of La Liga’s top stoppers.
With Costel Pantilimon having left the Etihad Stadium on a free number one Joe Hart is short on back-up, and Pellegrini sees his ex-star as a good option to keep the Englishman on his toes.
Caballero has shown reservations about taking up a role on the bench at City, but having missed out on a spot in his nation’s World Cup squad despite his fine displays, the 32-year-old may alter his stance.
The South American has two years remaining on his contract with the Spanish coastal side, but with their financial difficulties a sale may be sanctioned.
City have already added Fernando of Porto to their ranks and completed a free swoop for Bacary Sagna.
Alexis Sanchez of Barcelona is rumoured to be one of their next major targets, with the attacker having impressed at the World Cup with Chile.