The attacking midfielder was forced to leave the pitch after 70 minutes of play on his Milan return and underwent further examination on Monday to clarify the extent of his injury.
"AC Milan communicate that medical examinations have revealed an injury to Kaka's left adductor," a Milan statement reads.
"The player will undergo another series of tests in 10 days before making a prognosis about the date of return."
Kaka, meanwhile, has made it clear that he will waive his right to receive his wages as long as he is unavailable due to injury.
"Saturday was a bad day for me as I didn't expect this," the 31-year-old told Milan Channel.
"I've talked about this with the club and with [CEO] Adriano Galliani. What I have decided is that I don't want the club to pay me while I'm unavailable. All I want is the club's support and affection during my absence."
Milan is being plagued by injury at the moment, with Giampaolo Pazzini, Daniele Bonera, M'Baye Niang, Riccardo Montolivo, Mattia De Sciglio, Ignazio Abate and Stephan El Shaarawy also sidelined for this week's Champions League clash against Celtic.