The French forward has been heavily linked with a move away from the Spanish capital club after struggling for form this season and the Catalan club recently confirmed that they had approached Griezmann over a potential move to the Camp Nou.
The 26-year-old came close to leaving Atletico in the summer with United the favourites to land him, but decided to remain at the Wanda Metropolitano for at least another season after the club's transfer ban was upheld by the Court of Arbitration for Sport (CAS).
The club's ban ends in January, when their summer acquisitions Diego Costa and Vitolo will be eligible to play. Griezmann has a €100m (£88.7m, $118.4m) release clause which will become active in the New Year and Barcelona can make a move especially since they will have no competition from United, who were the front runners.
According to Spanish publication Marca, Jose Mourinho is behind the 20-time English champions' reluctance to re-enter a bidding war as he does not believe the France international is worth his hefty release clause. The Portuguese remains a fan of the player, but will not make a move to sign him in January or next summer.
United's exit from the race makes Barcelona the favourites to land the forward, who is expected to leave Atletico before the start of next season. The Catalans, however, are in a spot of bother in their pursuit of Griezmann after Diego Simeone's team confirmed that they had reported Barcelona to Fifa for approaching the player without permission.
"Atletico Madrid have presented a complaint about Barcelona for repeated contact between Barcelona, the player and his people," a statement from the club read.
"Griezmann has a long contract and this cannot be permitted, while the club also considers it could affect the competition [La Liga] in which Barcelona are currently top and Atletico are second."
Ernesto Valverde is keen to strengthen his offensive line and they have also been heavily linked with a move for Philippe Coutinho, who they view as a long-term replacement for Andres Iniesta. Barcelona made a number of offers for the midfielder last summer, but Liverpool stood firm and refused to sanction the sale.
The Camp Nou outfit have not given up their chase and are expected to return with another bid during the January transfer window. The Marca report claims that Barcelona are unlikely to sign both Coutinho and Griezmann and that club president Josep Maria Bartomeu will have to make a decision with regards to which player they want to sign.
The combined cost for both the players will be over €200m which even Barcelona are unlikely to be able to afford in a single window. The only way it is possible, according to the report, is if Valverde manages to offload a number of players currently out of favour at a premium.