The Cote d'Ivoire worldwide injured a ligament in a collision with Blues midfielder Matteo Kovacic during the second half.
Manchester United confirmed that Bailly's season is all but over and he will miss the final two encounters against Huddersfield Town and Cardiff City.
Bailly secured just his eighth start in the Premier League this season but the African centre-half lasted 71 minutes before he succumbed to injury.
The Africa Cup of Nations is from 21 June to 19 July in Egypt.
Ivory Coast are to play South Africa, Morocco and Namibia in the group stage, and Bailly would have been a certainty to be involved had he not sustained this injury.
Manager Ole Gunnar Solskjaer said after the game Bailly would nearly certainly miss the season's final games against Huddersfield and Cardiff and now his hopes of representing his country this summer are also over.
Victor Lindelof and Marcos Rojo are the only fit defenders in the squad who can occupy the central positions.
Meanwhile, United striker Romelu Lukaku has acknowledged David de Gea's poor run of form after the latest error from the goalkeeper in the Chelsea game led to the concession of another goal.