But with recent reports suggesting that the Frenchman is keen on remaining in the Spanish capital, United may have to find an alternative to the player they've had their sights set on for quite some time.
Griezmann is right up there with the world's best players; hence it will prove fairly difficult for the Red Devils to acquire a similar player in the summer transfer window.
Nevertheless, here are 5 players United could look to recruit if they ultimately fail to lure Griezmann to Old Trafford:
Lukaku's decision to reject a contract offer from Everton has undoubtedly put clubs in and beyond the Premier League on high alert.
He's evidently a quality striker, one that could be plying his trade at Old Trafford come next season.
United have already expressed some interest in signing Lukaku, but his former club Chelsea are said to be the frontrunners for his signature.
Jose Mourinho is looking to restore United to the pinnacle of English football next term and beyond and Lukaku could help them do just that.
It's only a matter of time before Lacazette makes his way to one of Europe's biggest clubs.
He's made a name for himself in Ligue 1 having scored bucketloads of goals for Lyon.
With 22 goals in 24 league appearances this term, it's hard to doubt Lacazette's quality especially doubled with the fact that he's been banging in the goals on a consistent basis for Lyon over the past four years.
The likes of Borussia Dortmund, PSG, Arsenal and United's fierce rivals Manchester City are rumoured to be interested in acquiring Lacazette's services; thus, the Red Devils are going to have to fight hard to secure the Frenchman's signature.
Dortmund's season has been full of ups and downs, succumbing to defeats in low-key games this term.
There's an apparent lack of ambition that has loomed over BVB in recent years as they've become accustomed to being second best in the German top flight since Jurgen Klopp's glory days.
Aubameyang is one of the most sought-after players in the world at the moment and with Dortmund seemingly stagnating, he's in need of a move to a side with genuine title hopes both domestically and in Europe.
Whilst United may not be title contenders just yet, they're poised to return to former glories in the coming seasons and the 27-year-old Gabon international could very well be at the forefront of that expected success.
He's scored 22 Bundesliga goals in as many appearances, for a total of 29 from 32, thus highlighting his prolificacy.
The 23-year-old has scored 23 league goals from 83 shots this season, thus highlighting his efficiency in front of goal.
He's definitely a player United should attempt to sign if the Griezmann deal goes south, although he does have a buyout clause of €100 million.
Muller's versatility up front would serve him well in England.
The World Cup winner is able to take up a number of roles in the attacking third including that of an out-and-out striker.
With just one league goal to his name this term, it's fair to say that he's had a relatively poor season, but a move to United could see him rekindle his career under Mourinho's guidance.