The Catalan giants squandered a 4-1 first-leg lead as they succumbed to three unanswered goals in Rome as the Serie A side progressed to the semi-finals on the away goals rule.
Barca have since confirmed Rakitic suffered a broken finger during the tie and was set to undergo surgery on Wednesday evening.
The news means the Croatia international is set to miss out when Barcelona look to return their focus to their LaLiga title charge at home against Valencia on Saturday.
Ernesto Valverde's men currently sit 11 points clear at the top of the Spanish Primera Division and are six games away from completing an undefeated march to the title.
However, Barca face a Valencia outfit who are just three points adrift of second-placed Valencia, and any slip-up at the Camp Nou this weekend could leave the European giants sweating over the final six matches of the campaign.