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.
Sky Sports reporter Dharmesh Sheth also opened up on United's Godin pursuit.
And with another potential target, Yerry Mina, on the verge of finalizing a move to Everton, United made an approach to sign the experienced Godin from Atleti on the deadline day.
The news comes just hours after manager Jose Mourinho outlined that the club would be making no additions to the squad on transfer deadline day.
The 32-year-old was reportedly offered a deal worth £8 million-a-year at Old Trafford but Godin turned down the opportunity in favour of a new contract at Atletico.
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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.
"So it's time, at least for me, to stop thinking about the market because the market will be closed".
"I've been told in the past few minutes Godin's agent was actually the one who approached United this week, saying the player was keen to come to Old Trafford and a deal could be done".
When asked about potential transfers at Thursday morning's press conference, Mourinho said he was "not confident" of bringing anyone in before the 5pm deadline.
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