The Ivory Coast defender was forced off with a problem on 52 minutes and was replaced by Marcos Rojo.
The Reds boss confirmed afterwards that Bailly has a knee issue which could be serious.
"He is injured and I am afraid he is badly injured," Mourinho told reporters at the post-match press conference. "In his knee, in the ligament area. He feels it is really bad."
Marouane Fellaini also appeared to pick up a knock following a high tackle from David Luiz just before half-time which earned the Chelsea defender a booking. The United midfielder did not return to the fray for the second half, with Juan Mata coming on in his place.
Wayne Rooney was absent from Sunday's match squad due to an injury he suffered in the Reds' last training session. The club captain had felt a problem during the Fenerbahce game but played on in his determination to complete 90 minutes.