The Uruguayan defender would be an alternative to linked names Toby Alderweireld, Jerome Boateng and Harry Maguire.
Jose Mourinho is desperate to land a central defender before the window closes on Thursday and had hoped that the 32-year-old Godin, who was named in the 2018 World Cup dream team after leading Uruguay to the quarter-finals, could be the answer to his problems.
A leader and a champion defender, Godin has a huge admirer in Manchester United manager Jose Mourinho.
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Mourinho hoped to trigger Godin's release clause - believed to be around £18m - and bring him to Old Trafford on an £8m a year deal in time to play against Leicester on Friday.
Godin has reportedly rejected a €9m-per-season offered by United in order to stay at Madrid.
According to Sky Sports, United have made a move for Godin and are chasing a deal for the 32-year-old, considered one of the finest defenders on the planet.
According to Sky Sports, United are chasing a deal for the Uruguay global.
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"I'm not confident, I'm not confident and the market closes today so is time at least for me to stop thinking about the market because the market will be closed".
It leaves United going back to the drawing board as they seek a defensive signing today.
But Mourinho told reporters at the beginning of deadline day that he didn't anticipate any new arrivals at Old Trafford.
United have signed three players this transfer window, Diogo Dalot, Fred, and Lee Grant.
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'I will have to focus on the players I have in relation to this first couple of matches.