The Uruguayan defender would be an alternative to linked names Toby Alderweireld, Jerome Boateng and Harry Maguire.
Mourinho downplayed United's chances of signing another player with just hours remaining of the transfer window.
Manchester United have made an offer to sign Diego Godin, according to Sky Sports.
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The 32-year-old has been one of the best central defenders in world football since joining Atletico from Villarreal in 2010 and he was once again a star performer for Uruguay at the World Cup this summer.
However, like the rest of their defensive targets, the Uruguay worldwide has proven to be out of reach.
And BBC Sport journalist Stone has now revealed that the Red Devils were always wary of bidding for the 32-year-old.
He said: "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".
A leader and tough uncompromising defender, he is a player Mourinho has coveted for some time and is a key component to Atletico where he has a strong relationship with coach Simeone and that relationship has led to a deadline day rejection.
"Behind the scenes, there was a suspicion they were being used as leverage by Godin's representatives to get a new deal from Atletico".
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.
On Thursday morning, several reports claimed that United had contacted Atletico in the hopes of adding Godin to their squad before the 5pm deadline.
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