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.
Mourinho downplayed United's chances of signing another player with just hours remaining of the transfer window.
Although Godin has a €20 million (£18m/$23.2m) release clause in his current Atletico contract he is happy to stay with Diego Simeone's side beyond the expiry of the existing deal.
United made contact with Atletico this morning about the veteran's availability, but the enquiry was quickly rebuffed.
"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", he told reporters when asked about his transfer options.
It's clear that United have been looking at adding a top quality centre-back and Godin has been one of the finest in Europe over the last number of seasons.
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"Behind the scenes, there was a suspicion they were being used as leverage by Godin's representatives to get a new deal from Atletico".