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The club revealed that both the 35-year-old Swedish striker and Argentinian defender Marco Rojo had suffered knee injuries in the same match.
"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," read the United statement on their website.
"Estimations of time to full recovery will only be possible once definitive treatment plans have been decided after these consultations."
Ibrahimovic didn't have his finest game agianst Anderlecht -- Marcus Rashford edging United through to the semi-finals on aggregate after extra-time -- but his goals this season have been the main reason for an improvement in the club's fortunes, albeit without ever being title challengers.
Ibrahimovic has defied many doubters since joining on a free transfer from Paris Saint-Germain last year, scoring 28 goals and inspiring United to victory in the League Cup.
Rojo's injury is also a serious bodyblow to United's depleted resources at the back with both England international central defenders Phil Jones and Chris Smalling expected to be out till mid May.