The Uruguay international was hit with the lengthy suspension by FIFA after biting the shoulder of Italy defender Giorgio Chiellini at the World Cup in Brazil.
The 27-year-old attempted to halve his punishment, but CAS decided to uphold the ban while allowing Suarez to train with his teammates and complete in friendlies.
"I'm very happy to feel like a footballer again and to be with my teammates," Suarez told the club's official website. "I was really looking forward to starting. The situation made me feel very uncomfortable.
"I'm paying for an error I committed and have apologised for, but it needs to be forgotten about. Now I have to think about the future, which is Barcelona, the club I dreamed about playing for."
Suarez, who joined Barcelona from Liverpool earlier this summer for a reported £75m fee, trained with his teammates for the first time today.