According to Miguel Delaney of The Independent, "senior figures" at the Manchester and Madrid sides believe the Frenchman is edging closer to the Camp Nou, with Barcelona "very far ahead" in the race for the player.
In addition, the report also stated that some Atletico players are growing more sure that Griezmann, who is valued in excess of £100 million, will join the current La Liga leaders at the end of the campaign.
"That is not particularly seen as a concern as United are considering a few options for the summer, in different positions," continued Delaney.
Per the report, United appeared to be set to land Griezmann in the summer, although the player decided to stay and signed a new contract when Atletico were hit with a ban on recruitment by FIFA.
For the Red Devils, it was a blow to miss out on Griezmann, not only because he would improve their attacking threat, but because he's developed into one of the standout footballers in the game. As noted by French football writer Jeremy Smith, he recently achieved a respectable ranking in the Ballon d'Or:
Griezmann has made major progress since arriving at Atletico Madrid in 2014 and linking up with manager Diego Simeone.
The talent and flair has always been obvious with Griezmann, although during his time at Real Sociedad he was unable to channel his quality consistently. In the capital that's changed dramatically.
No longer a wide player, the Atletico No. 7 is at the hub of his side's attack. At his best when drifting behind a centre-forward, Griezmann can dribble, pass and shoot, offering his side a varied and potent threat in the final third.
As noted by Spanish football journalist David Cartlidge after Atleti's elimination from the UEFA Champions League, so often this campaign the forward has been poor:
However, as relayed by OptaJean, after some early struggles, there were signs of Griezmann getting back to a level nearing his best prior to the recent knock that ruled him out of the game against Real Betis:
If United were able to sign Griezmann it would be a big boost, as Jose Mourinho's team would benefit from another attacking edge.
At times this season they've struggled without Paul Pogba in the side, with the midfielder adept at knitting passages of play together. While there are big differences in playing style and position, Griezmann would take some of the burden off his compatriot and add another goal threat.
The chance to move to Barcelona and play alongside the likes of Lionel Messi and Luis Suarez would also be a major temptation for any footballer. Having shone in La Liga for so long without actually winning the prize, Griezmann may also want some silverware to show for what has been a great spell in Spain.