Rojo left the field on a stretcher in the first half of United's win over Anderlecht in the Europa League on Thursday while Ibrahimovic sustained damage to his right knee late in the same game.
"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," United said in a statement on their website.
"Estimations of time to full recovery will only be possible once definitive treatment plans have been decided after these consultations."
The 35-year-old Ibrahimovic is United's top scorer this season with 28 goals in all competitions while Rojo has established himself at the heart of the defence.
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The loss of Rojo leaves manager Jose Mourinho short of cover at the back as the Argentina international's fellow central defenders Phil Jones and Chris Smalling are also injured.
United skipper Wayne Rooney, who has not played for three weeks due to an ankle injury, is back to full fitness and could replace Ibrahimovic for the rest of the campaign.
United play Spanish side Celta Vigo in the Europa League semi-finals next month and are fifth in the Premier League ahead of their next game at Burnley on Sunday.
(Reporting by Simon Jennings in Bengaluru, editing by Ed Osmond)
