It is believed that Manchester United were willing to pay off the 32-year old's £17.5 million release clause and had tabled a lucrative two-year contract worth £150,000-a-week in wages.
A leader and a champion defender, Godin has a huge admirer in Manchester United manager Jose Mourinho.
The newspaper also claimed United have contacted Atletico, with Mourinho determined to bring in one more signing.
The Red Devils flew out of the blocks with the purchase of Brazil midfielder Fred, but the subsequent captures of Diogo Dalot and Lee Grant have done little to inspire at Old Trafford.
Shortly after, though, and the rumours simmered down with reports that Godin had personally rejected a move and that he had in fact signed a new contract in Spain.
Having signed for the La Liga outfit in 2010, Godin has no intention of leaving this summer and, at the time of writing, he is close to committing his future to Diego Simeone's side.
United are in the market for a new centre-back but have faced frustration over their top targets.
Manchester United start their Premier League campaign tomorrow with a game against Leicester City.
The 55-year-old said: "The market closes today".