However, it appears the Frenchman's list of potential suitors is set to broaden with Spanish giants Real Madrid keen to join the race and land the talented midfielder as early as January, according to talkSPORT.
Reports suggest that Rafael Benitez wants to emulate Greizmann and Karim Benzema's partnership for the France national team at the Bernabeu, and believes the move could be the added impetus needed in their pursuit of La Liga and European triumph.
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The 24-year-old had a hugely successful debut season with Atletico Madrid, which prompted Premier League champions Chelsea to make a move over the summer - they failed in their pursuit, though.
Los Blancos will not be the only team in pursuit of the attacking midfielder come January, with Louis van Gaal's Red Devils similarly interested in forging a partnership that involves Griezmann - with summer signing Anthony Martial.
The Telegraph reports that United chairman Ed Woodward has told a fan forum: “The club will look to strengthen the squad and reward young talent.”
However, it is unlikely that Atletico will let their star leave on the cheap - if at all - given that Griezmann has a release clause worth somewhere close to £50 million.
United, though, would not shy away from such a fee, with it likely to be a case of whether Atletico's prized asset would be willing to make a move to Manchester.
The Frenchman has scored five goals in as many games in all competitions this term and will be largely accountable for the success of Atletico this season, with Diego Simeone's side currently fifth and five points off leaders Barcelona.