Rojo was stretchered off in the first half but his problem appears less serious than the injury sustained by Manchester United's top scorer in the final minute of normal time.
Reports suggest Ibrahimovic will miss rest of the season with knee ligament damage and if it is deemed serious enough to keep him out for several months he could already have played his last match for the club.
Jose Mourinho can recall Ander Herrera, Ashley Young, Matteo Darmian and David de Gea to his line-up at Turf Moor, while Wayne Rooney is also likely to feature.
"We are in trouble," said Mourinho. "In this moment, Eric Bailly played six matches in a row, and we lost Marcos Rojo. We don't have [Phil] Jones or [Chris] Smalling.
"We are in trouble and now we lose Zlatan too, but in attack, we have options and Rooney is coming, and let’s see if (Juan) Mata can also return before the end of the season. Let's see what happens.
"Herrera plays (at Burnley). He didn't play (against Anderlecht). One yellow he would miss semi-final. I was thinking about (bringing on) him and (Marouane) Fellaini both at the same time, but because I did a change with Marcos Rojo I couldn't.
"He didn't play so Herrera is fresh to play, Ashley Young is fresh to play and who else? (There is) nobody else to play."