Chelsea have been searching for a left back to replace Ashley Cole for several seasons, with Filipe Luis returning to Athletico Madrid, and Baba Rahman currently the only recognised left back in the squad. Azpilacueta has been featuring at left back for a long time, and Mourinho would ideally move him to right back, where Branislav Ivanovic has been struggling so far this season.
Man City's left backs are Gael Clichy and Aleksandar Kolarov, who are 30 and 29 respectively, and both Pellegrini and Mourinho have been looking at Valencia left back, Jose Gaya.
21 year old Gaya has featured for Spain U21's and is seen as a future full international, and with Real Madrid rumoured to be interested, there could be a full scale bidding war for his services.
The story is reported in the Metro: http://metro.co.uk/2015/09/15/chelsea-set-for-transfer-battle-with-manchester-city-and-real-madrid-to-sign-valencia-defender-jose-gaya-report-5391519/
Chelsea are rumoured to be in the market for several players in January after a shaky start to their Premier League title defence (the worst by any side since Blackburn in 1995). While Manchester City are still seeking performances in the Champions League after another dissapointing night at the Etihad as they slipped to a 2-1 defeat to Juventus.