The Premier League Champions see the young starlet as a long term replacement for Joleon Lecott who will see his contract expire at the end of the season, becoming part of the expected mass exodus at the Etihad in the summer.
The 19-year-old has been catching the eye of top clubs across Europe, including last week’s performances in the Copa del Rey semi-final first leg against Barcelona, where he put in an impressive display including a late equaliser.
City will face stiff competition from free spending Paris Saint-Germain, who are also looking to strengthen their defensive options and could provide extra pull for the former Lens defender with a return to his native country, after Carlo Ancelotti also expressed an interest.
Varane has made 14 appearances for Los Blancos this season and there is still every chance he will stay in Madrid.
Signed in 2011 for a fee believed to be in the region of £9millon means it unlikely the Spanish champions will want him to leave, especially as he is starting to find his feet.
Both PSG and Manchester City have demonstrated that they are not scared of spending audacious amounts of money, if the right bid is received in Madrid it might persuade them to say adios to the youngster.