The Red Devils were believed to be interested in signing Griezmann earlier on in the summer but with Atletico not willing to sell, the Premier League giants recruited Romelu Lukaku instead from Everton for around £75million.
Atletico can also return to the transfer market from January, enabling the club to identify and pursue potential squad reinforcements in the event of Griezmann leaving the club next summer.
The forward revealed he did not want to leave the club while a replacement could not be found - saying it would be a "dirty move" to leave.
The Spanish side are banned from registering any players until January 2018, and Mourinho is already planning to relaunch the club's pursuit at the end of the season.
Griezmann, 26, had been a top target for Mourinho before the Special One changed tack and prioritised a No 9.
Griezmann has a €100m (£87.8m) release clause - which now seems cheap after the whirlwind final months of the summer - that Mourinho would be happy to trigger.
Sources at United were careful to insist in the summer that their interest in Griezmann was shelved rather than ended, with the intention being to leave the door open for a renewed move without antagonising the player.
With the ban being lifted in January and Atletico re-signing Diego Costa from Chelsea, Mourinho feels that this is the flawless opportunity to land his No.1 transfer target.
The Blues lost their opening two games of the season but have gone on to win six of their last seven, including a 4-0 thrashing of Stoke City over the weekend.