Sky Sports News understands that United have approached Atletico Madrid with a view to signing the Uruguay defender, as Jose Mourinho is desperate to strengthen at the back before the transfer window closes.
The 32-year old is widely regarded as one of the best central defenders in the game and has been an integral member of Diego Simeone's Atletico Madrid for years now.
Supporters are starting to lose faith in Phil Jones and Chris Smalling, Eric Bailly fell out of favour at the back-end of last season and Victor Lindelof has struggled in the Premier League.
But United appear to still be trying to bring in a top defender with attention suddenly turning to Godin.
"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".
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The report claims United also offered to double Godin's wages to £8.1m-per-season after tax but unfortunately they have not been given any encouragement that a deal can be done before today's 5pm deadline.
"So I will have to focus on the players I have".
"The information I have is no", said Mourinho when asked if he expected any signings or departures from Old Trafford on deadline day.
The Portuguese is desperately seeking reinforcements for his defence on the eve of the campaign with summer-long interest in Toby Alderweireld and Harry Maguire failing to result in moves ahead of deadline day.
Obviously Godin would be a superb short-term addition if United were able to pull it off but it's looking highly unlikely unless things take a drastic change over the next couple of hours.