The 32-year-old has plied his trade in the Spanish capital since 2010 following a switch from Villarreal.
Ahead of Thursday evening's transfer deadline the Red Devils have made various advances for Harry Maguire, Yerry Mina and Jerome Boateng as back-up options should they not push the Alderweireld deal over the line.
However, like the rest of their defensive targets, the Uruguay worldwide has proven to be out of reach.
Manchester United executive vice chairman Ed Woodward and head of corporate development Matt Judge are both at the club's Carrington training complex this morning as they try to deliver a late signing for Jose Mourinho.More news: Blues Set to Break World Record for Courtois Replacement
United have signed three players this summer, with Diogo Dalot, Fred and Lee Grant being added to the squad.
"It is understood United were encouraged to contact Atletico to see if there was any chance of a deal being done".
"Those suspicions were confirmed once talks started".More news: I don't need to explain or defend my relationships: Priyanka Chopra
According to Sky Sports, United are chasing a deal for the Uruguay worldwide.
'I will have to focus on the players I have in relation to this first couple of matches.
"It's my information United made an offer for the agent to take to the Atletico board".
However, although no there has been official confirmation of a new contract, United were told that Godin has agreed a new contract - and would remain at the Wanda Metropolitano with Diego Simeone's side for at least one more campaign.More news: Everton on threshold of signing Bernard from Shakhtar Donetsk on free transfer
"We should make it clear here, Mourinho would have been keen to sign Godin".