The 32-year-old has plied his trade in the Spanish capital since 2010 following a switch from Villarreal.
United then tried to peruse, Bayern's Jerome Boateng, Barca's Yerry Mina, and Tottenham's Toby Alderweireld but could not finalize the move with any one of them.
However, like the rest of their defensive targets, the Uruguay global has proven to be out of reach.
A leader and a champion defender, Godin has a huge admirer in Manchester United manager Jose Mourinho.
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The Portuguese has made three signings this summer - Fred, Diogo Dalot and Lee Grant - but has so far been frustrated in his search for a centre-back.
United were hopeful that they could have persuaded Godin to sign and join his new teammates in time for the Red Devils' Premier League season opener against Leicester on Friday night but they will have to return to the drawing board now.
"Those suspicions were confirmed once talks started".
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The Portuguese coach said on Sunday his team could face a "difficult" season if there are no new arrivals before the 5 p.m. deadline on Thursday.
Independent.ie understands that the approach for the player was speculative from United, with suggestions that the Old Trafford club could trigger a release clause in Godin's contract irrelevant as the player has made it clear he has no intention of leaving Atletico Madrid this summer.
"So I will have to focus on the players I have and in relation to this first couple of matches, maybe four matches because after Burnley there is national team and a couple of weeks to recover people and prepare people better, is to focus on players available so I'm not thinking about that list I give you, I focus on what we have".
Godin has reportedly rejected a €9m-per-season offered by United in order to stay at Madrid.
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"But he believes the desire of the agent's part was always to leverage the situation to get a new contract for his client".