The Ivorian has been an ever-present this season, starting all 12 of United's Premier League and Europa League games
"I wanted to thank you for all the messages of support that I received since the injury!" the Ivory Coast defender wrote on his official Facebook page.
"I hope to be able to play again for United and The Elephants before 2 months, God willing. Thanks for always being there!"
The message was accompanied by a picture of Bailly sitting in the passenger seat of a car at United's training ground, with his right leg in a brace.
A close-season signing from Villarreal, Bailly limped off early in the second half at Stamford Bridge on Sunday, with Marcos Rojo coming on to take his place.
"He is injured and I'm afraid he's badly injured," manager Jose Mourinho said in his post-match press conference.
"In his knee, in the ligament area, he feels that is really bad."
Bailly, 22, joined United from Spanish side Villarreal in June for a reported fee of 30 million pounds ($36.7 million, 33.7 million euros).
The Ivorian has been a virtual ever-present this season, starting all 12 of United's Premier League and Europa League games and only sitting out a League Cup tie.