Mandzukic out for up to two weeks
The Croatia international striker suffered the injury after being elbowed in the faced by Olympiacos' Pajtim Kasami in the opening stages of Tuesday's UEFA Champions League contest.
Mandzukic was able to play on for the entire match and found the net as Atletico were beaten 3-2 in their Group A opener.
Atletico announced on Wednesday that the forward was to have surgery, and an update was provided by club doctor Pedro Luis Llopis on Friday.
"He's hoping to return to training on Sunday or Monday," the doctor explained in quotes reported by AS.
"He trained the other day, before undergoing surgery which suggests how keen he is to get back with the squad.
"Under normal circumstances, a player with an injury like this would hope to be back playing in 15 days but if he wears a protective mask, he should be able to play within 10 days.
"He will be issued a personalised mask, the smallest and most comfortable one possible, although it's always a little complicated to play whilst wearing one."
Mandzukic's estimated period of absence means he is scheduled to miss three La Liga games, with the Champions League match at home to Juventus on October 1 earmarked as a possible return date.