Jose Mourinho confirmed Eric Bailly sustained a knee ligament injury during Manchester United's 4-0 defeat to Chelsea on Sunday.
Bailly collided with Eden Hazard early in the second half and hobbled off to be replaced by Marcos Rojo in the 52nd minute.
Mourinho fears it could be a "bad" injury, with the Ivorian centre-back now facing a lengthy spell on the sidelines.
"He's injured and I'm afraid he's badly injured. In his knee, in the ligament area, he feels it is really bad," Mourinho told a post-match news conference.
United face Burnley and Swansea City in their next Premier League outings and the Portuguese expects his team to pick up points and remain close to the leading trio of Manchester City, Arsenal and Liverpool, who they sit three points adrift of.
He added: "We are six points from the top of the league, not 16 points. I don't know but three, maybe four from the top four, so we are there with very difficult fixtures.
"I am not saying the matches that are coming are easy ones, far from it, but we have to beat opponents in mid-table that we have now - Burnley, Swansea, Sunderland, Middlesbrough. They are difficult but we need points from the upcoming matches.
"The top teams have to play each other so they have to lose points, so in the end we will be around."
Mourinho exchanged words with Antonio Conte following the final whistle and appeared to be disgruntled by the Italian encouraging the home fans to cheer after N'Golo Kante made it 4-0.
However, the United boss refused to reveal the specifics of their conversation.
"I am not the kind of guy to come here and share with you things between me and Antonio. It stays between me and him, unless he wants to share it," he said.