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.
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.
His loss is a blow to United's hopes of European success and their chances of securing a top-four finish in the Premier League.
At times he has carried the team this season, his form earning him a PFA Player of the Year nomination.
United travel to Burnley on Sunday knowing victory will see them cut the gap to fourth-placed Manchester City to one point, with the city rivals going head-to-head at the Etihad Stadium on Thursday.
The Europa League offers an even stronger chance of Champions League qualification, though, with only two legs against Celta Vigo and a final against either Lyon or Ajax standing between United and a maiden triumph in the competition, and with it a place at Europe's top table next season.
Ander Herrera and Ashley Young, described by Mourinho as “fresh to play” after not featuring in midweek, appear set to feature at Turf Moor, where Burnley have won 10 of their 16 league games this season.