The 29-year-old has proven pivotal in helping Atletico achieve their recent domestic success - winning La Liga in 2014 - and has attracted interest from a host of clubs across Europe.
And while the Spanish side would rather retain the services of their star centre-back, they could be ultimately rendered powerless should City meet his release clause and the Argentinian decide on a new career path.
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After signing starlet Jason Denayer on a new five-year contract, the Citizens seemed willing to give the 19-year-old a shot to earn first-team football at the beginning of the season, although it would now appear that City would prefer to move for a world-class central defender.
The heart of City's defence was the focus of much criticism last season with Vincent Kompany's niggling injuries, Martin Demichelis' lack of pace and newboy Eliaquim Mangala's struggle to find his feet in England resulting in inconsistent partnerships.
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City would face a challenge for Godin's signature from rivals Manchester United, who are similarly on the look out to improve their already-fragile defence.
Manuel Pellegrini's side have also been heavily linked with Wolfsburg midfielder Kevin De Bruyne, but his £50 million price tag has proven a sticking point.