The Uruguayan defender would be an alternative to linked names Toby Alderweireld, Jerome Boateng and Harry Maguire.
Godin, 32, has been one of the world's most consistent defenders over the past five years.
Signing Godin at this late stage would be incredible business by United, but it is also highly difficult.
Atletico have resisted offers for Godin in previous summers and it would have to be a substantial bid to convince them to sell at this late stage.
Godin would have to agree personal terms and a work permit, while a formality, would still have to be processed before today's 5pm deadline.
United have signed three players this transfer window, Diogo Dalot, Fred, and Lee Grant.
Jose Mourinho's side kick off their Premier League campaign this Friday against Leicester City.