City have found themselves linked with the Atletico Madrid defender for quite some time, whilst Chelsea and Liverpool were also credited with an interest in the summer.
City eventually bought in Martin Demichelis from Atletico, although it is understood Manuel Pellegrini is a big fan of Godin.
But according to reports in Germany, Borussia Dortmund are set to make a January offer for the Uruguayan defender, as they fear they could end up losing Neven Subotic to Manchester United.
Manager Jurgen Klopp is anxious to find replacements quickly, with the squad that reached the final of last seasons Champions League seemingly becoming thinner by the week.
After losing star man Mario Gotze in the summer, striker Robert Lewandowski confirmed this week he will be joining Bayern Munich at the end of the season and will sign a pre contract agreement in January.
Klopp knows he will need a top quality replacement should Subotic also depart, and reckons 27-year-old Godin fits the bill.
Godin has spent six years in Spain, and previously played for Villarreal.
His current contract at Atletico Madrid expires in 2015.