The duo both picked up the injuries during Thursday night's Europa League win over Anderlecht and, although United did not put a time frame on their recovery, look likely to miss the rest of the season at least.
A statement from the club said: "Detailed investigations on the injuries sustained by Marcos Rojo and Zlatan Ibrahimovic during Thursday's Europa League match have confirmed significant knee ligament damage in both players that requires specialist opinions over the coming days.
"Estimations of time to full recovery will only be possible once definitive treatment plans have been decided after these consultations."
The loss of Ibrahimovic is a blow to United's hopes of European success and their chances of securing a top-four finish in the Premier League.
The 35-year-old striker who has scored 28 goals in all competitions in his debut competition in English football, suffered his injury when his knee buckled in an awkward fall.
Rojo had left the pitch on a stretcher earlier in the match against the Belgians, which United won 2-1 after extra-time to advance to the semi-finals.
Ibrahimovic's one-year contract is up at the end of the season, meaning it is possible the Swede has played his last game for the club, with speculation linking him with a move to Major League Soccer.
At times he has carried the team this season, his form earning him a PFA Player of the Year nomination.
However, it is the loss of Rojo which is proving of greater immediate concern to United boss Jose Mourinho, who is having to cope with a debilitating injury list.
While the Portuguese is confident he has "options" up front, with Wayne Rooney ready to return, on the defensive front he is already without long-term absentees Phil Jones and Chris Smalling.
He said after the Anderlecht game his team were "in trouble" due to injuries.
EPL Table | Premier League standings after Gameweek 34