The 35-year-old was released by the Manchester club as he was no longer included in Pep Guardiola's plans for the new season at the Etihad. City released four players after the end of the season and the Argentine goalkeeper was one of those in the list, along with Bacary Sagna, Gael Clichy and Jesus Navas.
According to the Mirror, the Blues have held talks with Cabellero to become their second choice goalkeeper at Stamford Bridge. Antonio Conte had Thibaut Courtois as his first choice keeper in their title winning campaign, while Asmir Begovic was a back-up.
Begovic left the west London club and has already agreed to join AFC Bournemouth. The Bosnia and Herzegovina international's departure from the club has forced them to enter the market in search for a new back-up for Courtois.
Chelsea have identified Caballero, who will replace Begovic in their squad as Conte's side look to defender their title in the 2017/18 season. The capital club have already held talks with Guardiola's former player as the Blues manager wanted the club to fill in that role quickly, following Begovic's exit.
The move will not be made official anytime soon and it will be announced on 1 July. The Mirror has not mentioned the reason why there is delay in announcing Caballero's arrival.
Meanwhile, Conte is keen on securing the services of Everton striker Romelu Lukaku and AS Monaco midfielder Tiemoue Bakayoko. However, neither of the two will be the first addition of the summer transfer window. With Caballero set to become Chelsea's first signing, the rest are likely to follow him to Stamford Bridge.