The Portugal international is high on Manuel Pellegrini's wish-list - and the Chilean, who arrived at the Etihad Stadium earlier this summer, has stepped up his pursuit of the 30-year-old after Matija Nastasic was ruled out for the first few weeks of the season with an ankle injury.
However, securing the services of Pepe could prove difficult for the Premier League outfit.
Earlier this summer they had a £15m bid rejected by Los Blancos, whose new coach Carlo Ancelotti appears keen to keep the central defender at the Bernabeu.
Pepe joined Real Madrid in 2007 from Porto and has won two La Liga titles and a Copa del Rey during his six-year stint in the Spanish capital.
His temperament has often be called into question, though, while he spent some time on the Real Madrid bench last season following the emergence of 20-year-old France international Raphael Varane.
Manchester City Fixtures
Monday August 19
20:00 Premier League Manchester City v Newcastle United
Sunday August 25
16:00 Premier League Cardiff City v Manchester City
Saturday August 31
12:45 Premier League Manchester City v Hull City
Saturday September 14
15:00 Premier League Stoke City v Manchester City
Sunday September 22
16:00 Premier League Manchester City v Man Utd SS
Saturday September 28
15:00 Premier League Aston Villa v Manchester City