The Real Sociedad winger has emerged as one of European football’s top wide-men over the course of the last year after helping his side secure Champions League qualification for this season.
His performances have long been attracting attention from Arsenal, but it now appears that United are also keen.
Moyes is said to be desperate to add proven quality to his squad this summer, and has made a top class wide-man is a priority.
Griezmann is valued at around £25m, and is seen by the Red Devils as a realistic target, despite there being serious interest from a number of top European clubs – including PSG.
The 22-year-old is said to be open to a big move later this year, with Sociedad no longer seen as a club capable of matching his level of ambition.
United have had scouts assessing the Frenchman in recent months, and could dedicate a large chunk of their rumoured £100m transfer budget to securing a deal.
Griezmann is expected to feature at the World Cup, and could delay a decision on his future until after the tournament.