The Sky Blues have been in the market for a centre-back for some time, with the likes of Martin Demichelis and Matija Nastasic having failed to impress enough to be considered as an automatic starter alongside the club captain.
Eliaquim Mangala and Mehdi Benatia of Porto and Roma were rumoured to have been City’s main targets, but deals for either look to be difficult.
Such blows have forced Pellegrini to look elsewhere, and Pepe is believed to be a player he sees as ideal to work alongside Kompany.
The 31-year-old Portugal international has been a mainstay in Real’s backline for some time, but it looks likely that he will be eased out completely next term in favour of emerging Frenchman Raphael Varane.
Pellegrini worked with the experienced enforcer while at the Bernabeu, and is known to be a fan of his abilities.
A bid of around £25m is likely to be enough to get him across to the Etihad Stadium and should not make a major dent in City’s summer budget.