Gunners boss Arsene Wenger rates the France U21 midfielder highly.
The Daily Express says United boss David Moyes is also tracking Griezmann, who could cost as much as £20million.
A battle may ensue between Arsenal and United for the Frenchman, with Sociedad admitting that they may be forced to sell.
"We want to keep him for as long as possible," a club source told Marca. "But we have a battle to convince him to stay – we know clubs are interested in him."