Manager Jose Mourinho described the defender’s injury as “really bad” after Sunday’s 4-0 defeat to Chelsea at Stamford Bridge.
The 22-year-old, who has impressed at United since making a £30m move from Villarreal in the summer, hurt himself in a clash with Eden Hazard in the second half.
There has been no official word from United, but Bailly has taken to Instagram to give an update on the situation.
He wrote: “I wanted to thank you for all the messages of support that I received since the injury! I hope to be able to play again for United and The Elephants before 2 months, God willing. Thanks for always being there!”
United are currently seventh in the Premier League table, six points behind the top three in the standings.