The Catalan outfit are understood to be looking to offload the 30-year-old, who is now the club's highest earner after Lionel Messi officially became a free agent this month.
Barca president Joan Laporta have been instructed by La Liga chiefs to reduce their wage bill before the 2021-22 campaign begins in order to comply with financial fair play regulations.
The forward is also believed to have been offered to Premier League clubs, including Manchester City and Chelsea, and the Daily Mirror reports that United are the latest team that Barcelona have been in touch with.
A deal for Griezmann to join Ole Gunnar Solskjaer's side seems unlikely, however, as the Red Devils are not prepared to pay a major fee for the Frenchman, while they are also prioritising other targets, including Real Madrid's Raphael Varane. body check tags ::