The defender also called on FIFA to reduce the Uruguay striker's four-month suspension.
Suarez has been banned from all football-related activities, both internationally and at club level, for 16 weeks after he was found guilty of biting Chiellini during La Celeste's World Cup last 16 win over Italy last week.
The Liverpool striker has since admitted to the incident and apologised to his counterpart, who accepted the gesture before calling on world football's governing body to reconsider their ruling.
"It's all forgotten," the Italian wrote on Twitter. "I hope FIFA will reduce your suspension."
.@luis16suarez It's all forgotten. I hope FIFA will reduce your suspension.— Giorgio Chiellini (@chiellini) June 30, 2014