Manager Jose Mourinho is still looking to improve his side's attack in order to compete with the best in the Premier League and Europe, and the Portuguese sees the 25-year-old France forward as a player who can provide quality across a number of positions at Old Trafford.
Griezmann is under contract at Atletico until 2021 and has long maintained his happiness at the Vicente Calderon, saying last year that he would stay at the club as long as coach Diego Simeone is in charge.
The Argentine has cut his contract by two years to 2018 and is expected to join Inter after leading the Rojiblancos in their first year at their new stadium, the Wanda Metropolitano, in 2017-18.
But the high costs involved in the construction of that stadium mean Atletico are strapped for cash and the Madrid club will likely look to raise more funds in the summer through the sale of one of their big players - possibly Griezmann to United.
Atletico would prefer not to sell Griezmann, who signed a new deal last June, and club president Enrique Cerezo said recently: "I don't know what Manchester United's interest is. But Griezmann will be at Atletico for a long time."
Talk is cheap when it comes to transfers, but it is true that Atletico (currently unable to sign next summer due to a transfer ban) would rather part company with another of their players instead of losing their prized possession - a footballer who recently stood alongside Cristiano Ronaldo and Lionel Messi on the shortlist for the Ballon d'Or and FIFA's The Best prizes.
At the ceremony for The Best in January, Griezmann was asked if he would be leaving Atletico and said no. Nevertheless, the Old Trafford club are working hard behind the scenes to try and seal his signature.
When he renewed his contract last summer, the 25-year-old's release clause was set at a prohibitive €100m. However, if Atleti decide to sell, they could be tempted by an offer in the region of €80m, which would represent a €50m profit on a player bought for €30m from Real Sociedad in 2014.
Paul Pogba has been trying to persuade Griezmann to join him at United and the Atletico forward admitted in an interview in November that he speaks regularly to his friend and international team-mate about the club, Jose Mourinho and also the Premier League.
"I always ask Pogba about Manchester United," he said. "If some of the players are that good. If Mourinho is that good. It's an enormous club, a really good project. I have heard a lot of things about United and PSG, but at the moment I don't see myself at another club."
But Griezmann is clearly fond of United and admitted last year that his idol was David Beckham. Meanwhile, his brother also intensified speculation of a possible move when he wrote on Twitter following Memphis Depay's transfer to Lyon: "Sorry Memphis, I think Old Trafford is better!"
With Wayne Rooney set to move on in the summer, Griezmann is an interesting choice to revitalise the attack at United, with the France forward able to play across the front three, as a main striker or as a number 10.
Mourinho believes Pogba could be key to sealing his signature, but it may be more realistic for United to wait one more season because Griezmann is happy at Atletico right now and not convinced he wants to swap Spain (where he has lived since 2009) for England. He would also want to continue playing in the Champions League and right now, that is not something United can promise as they sit sixth in the Premier League.
Inter also hold an interest in Griezmann, but believe he will end up moving to United. However, Milan could interest the player if he wants to link up again with Simeone - a coach he admits has totally transformed his career - in the future.
Atletico are currently in fourth in La Liga, just a point clear of Griezmann's former club Real Sociedad. Failure to qualify for the Champions League could hasten his departure due to the player's own ambition and the loss of funds for the Rojiblancos if they miss out on Europe's premier club competition.
Otherwise, however, both he and Simeone could stay together for one more season and any interested clubs - United among them - would have to come back again in 2018.