According to the report, United will have to pay at least 100 million euros ($108.11 million) to secure the 25-year-old forward, who has scored 14 goals in 39 appearances for Les Bleus and was instrumental in the side's run to the Euro 2016 final.
Griezmann was the leading scorer in the tournament with six goals but hosts France lost 1-0 to Portugal in the title decider.
The Press Association, however, is reporting that no agreement has been reached, citing a Manchester United source.
News that makes sense
Your trusted source for staying up-to-date with the world around you. Get free daily news updates and analysis, straight to your inbox.
United are sixth in the Premier League, 14 points behind leaders Chelsea.
(Reporting by Julien Pretot; Editing by John O'Brien)
