The Uruguayan bit the arm of the Serbian during the second half of the 2-2 draw at Anfield and will be investigated by the Football Association for his actions which left Ivanovic fuming according to teammate Petr Cech.
"I didn't see it at the time but people have told me which is surprising," Cech told BBC Sport.
"Suarez always does little fouls and pushes. I saw Branislav's reaction and he was not happy at all.
"Every time Luis Suarez is close, he pushes defenders around. I was complaining to the referee about it."