According to Sport (h/t Sport Witness), while City have registered their interest in the France international, he favours a move to the Camp Nou. The report stated both Manchester clubs have made contact with Griezmann's representatives.
Additionally, a report from Le 10 Sport (h/t Sport) suggested the Blaugrana have already struck an agreement in principle that would see the player make the move to Catalonia next summer.
The Sport report added there's a release clause of €100 million (£89 million) in the forward's Atletico contract and Barcelona did attempt to sign him in the summer following the sale of Neymar to Paris Saint-Germain.
The move would be a fascinating one if it were to go ahead, as Griezmann has been one of the standout performers in Spanish football for a long time. As noted by OptaJose recently, typically his goals have been vital ones for Diego Simeone's side, too:
20 - Antoine Griezmann has scored more winning goals than any other player in La Liga since 2015/16. Star. https://t.co/c2Lrv1EFr72017-11-8 18:08:57
Since making the move to Atletico, the forward's game has gone from strength to strength, with his manager opting to use him through the middle as opposed to on the wing.
Griezmann relished the extra responsibility, and in a team that plays vertical football in attack, his ability to surge from deeper positions with and without the ball has caused massive problems for opposition defences.
Last term he was sensational overall, dropping into deeper pockets to create chances as well as scoring. But in 2017-18, after a summer of speculation about his future, he's not quite hit the same heights.
As Sky Sports Statto noted, his goal for France during the international break was overdue by his high standards:
Sky Sports Statto @SkySportsStatto
Antoine Griezmann scores his 1st goal in 7 apps for club & country - now scored 4 goals in last 5 caps at home for France https://t.co/vnUg3Hgz9j2017-11-10 20:25:27
Worryingly for Griezmann and Atletico, his last goal at club level was in September against Chelsea in the UEFA Champions League.
This mini drought is unlikely to detract potential suitors, though, as the Frenchman has proved himself over a long period of time. And for a Barcelona attack that has two key men over the age of 30 in Lionel Messi and Luis Suarez, a younger edge would be beneficial.
It's easy to see Griezmann fitting in at the Camp Nou. Not only is he used to the league, his versatility and dynamism mean he can be effective in so many different roles on the field.
However, as we can see courtesy of Goal UK, Griezmann's Atletico team-mate Koke believes he's happy where he is:
Goal UK @GoalUK
Koke is adamant Barcelona-linked Antoine Griezmann is happy at Atletico Madrid. 🔴 https://t.co/nZzIuv0EVt2017-11-16 17:23:18
If the Manchester clubs were to come in for the player, there would be a temptation to move to the Premier League. Both Pep Guardiola and Jose Mourinho are doing outstanding work at City and United, respectively, and they are managers with the quality to improve Griezmann's game even more.
But there's something special about Barcelona, and Griezmann will know that having spent such a long portion of his career in Spain. That's why it's easy to see him choosing the Blaugrana from the trio of options mentioned.