Man City were preparing a £28.1m bid for the 29-year-old defender, which would have activated his release clause.
However, Atletico Madrid have reportedly agreed to a contract extension with the Uruguay international.
Godin has also been pursued by Manchester United and Chelsea, who are also keen on signing defenders ahead of the new season.
Atletico Madrid manager Diego Simeone made it clear that he wants his star defender to remain in Spain, as he plans to raise Godin’s release clause to shoo away Man City and other potential suitors.
Chelsea are expected to sign Augsburg defender Baba Rahman for £17.5m after seemingly losing out on Everton defender John Stones.
Meanwhile, Man United have had no luck in upgrading their defense after signing former Torino defender Matteo Darmian.
The Red Devils are continuing their hunt for Real Madrid’s Sergio Ramos and Valencia’s Nicolas Otamendi, however.
United also reportedly remain in the race for Stones should he make the move away from Goodison Park.
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