The France international was heavily linked with a move to Old Trafford before the start of the 2017/18 season, with United manager Jose Mourinho keen on securing the forward's services. However, the Court of Arbitration for Sport (CAS) upheld Atletico's Fifa transfer embargo, forcing the player to continue at the Spanish capital club.
Griezmann signed a new contract at Atletico and his release clause will be reset at €100m (£88.3m, $117.5m) from the start of the next calendar year. But this has not stopped United's interest in the forward, while Barcelona have also joined the race for his signature.
However, according to the Independent, senior figures at United and Wanda Metropolitano believe the former Real Sociedad star will swap Atletico for Barcelona after the end of this campaign in a mega-money deal.
The report also claims the 20-time English champions' interest in the France international cooled down after they failed to sign him in the summer, and that United's hierarchy has already started looking for alternative targets for the next summer transfer window.
Earlier, his mentor and former agent Eric Olhats admitted the forward would have been at United in the summer, if not for Atletico's transfer ban. The player had also opened the door to joining the Premier League giants after confirming that a transfer was a "possibility".
However, when the prospect of signing the Atletico attacker came up once again recently, United were told that Barcelona are "very far ahead" in the race to secure the player's services.
His club teammates are also aware that the forward will leave Diego Simeone's side after this campaign. Despite the release clause, it is believed the La Liga leaders are ready to pay a fee well in excess of £100m for Griezmann.
Barcelona are looking to bolster their attack following Neymar's departure after the Brazil international completed a transfer to Paris Saint-Germain for a world-record fee of €222m (£200m, $264m) in the summer.
Ernesto Valverde's side signed Ousmane Dembele from Borussia Dortmund as a replacement, but failed to land Liverpool star Philippe Coutinho in the same window.
As for Coutinho, Barcelona have maintained their interested in signing the former Inter Milan man as he still remains a transfer target for the Blaugrana.