The Colombia striker joined Rudi Garcia's side on an initial loan deal on Sunday, with Roma able to make the move permanent for €12.5 million at the end of the campaign.
Ibarbo made his Roma bow as a substitute in Tuesday's 2-0 Coppa Italia quarter-final defeat to Fiorentina, but he will now be forced to spend a spell on the sidelines.
The news will come as a blow to Garcia, with Ibarbo arriving as Mattia Destro swapped Roma for Milan, while fellow January signing Seydou Doumbia is still on Africa Cup of Nations duty with the Ivory Coast.
A club statement read: "Ibarbo underwent medical tests today [Thursday] that confirmed a grade one/two injury to the myotendinous junction of his right medial gastrocnemius muscle.
"The player has already started treatment on the injury."
Ibarbo will watch on as Roma travel to his former club, Cagliari, on Sunday.
