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In an earlier statement, he said: "I apologise to my manager and everyone at Liverpool for letting them down."
Reds manager Brendan Rodgers added: "Having reviewed the video footage and spoken to Luis, his behaviour is unacceptable and I have made him aware of this."
Blues defender Ivanovic, 29, complained to referee Kevin Friend that he had been bitten on his right arm by Suarez after their clash inside the Chelsea penalty area.
Ian Ayre, the Premier League club's managing director, added Suarez's behaviour did not "befit" that of a Liverpool player.
"Luis is aware that he has let himself and everyone associated with the club down," said Ayre, who cancelled a planned trip to Australia to personally handle the fallout.
"We will deal with the matter internally and await any action from the Football Association."
Shortly after the match, Rodgers had said no individual was "bigger than the club" after pledging to review the incident.
"This is a club with incredible values and ethics," he said. "There's certainly no-one bigger than this club, a player or manager.
"As football managers, staff and players, we're representing this club, off the field and in particular on the field."
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