The Barcelona striker was suspended from all football-related activity until the end of October by FIFA in June after biting Giorgio Chiellini during Uruguay's World Cup clash with Italy.
Suarez, who joined Barca in a deal reported to be worth €94 million (£75m) from Liverpool last month, was also banned for Uruguay's next nine competitive international matches.
The 27-year-old's initial appeal was rejected by world governing body FIFA, so the case was taken CAS.
Legal representatives of Suarez were reported on Thursday as saying that they were "confident" of having the ban reduced.
And CAS revealed on Friday that they should be in a position to deliver a verdict by the end of next week.
A CAS statement read: "The Court of Arbitration for Sport (CAS) has today heard the appeal of Luis Suarez, FC Barcelona and the Uruguayan FA against FIFA. The hearing took place at the CAS headquarters in Lausanne, Switzerland.
"The player was present and gave a statement to the panel of CAS arbitrators: Mr Bernhard Welten, Switzerland (President), Professor Luigi Fumagalli, Italy, and Dr Marco Balmelli, Switzerland.
"At the end of the hearing, the panel informed the parties that it will issue its decision as soon as possible, probably before the end of next week."
The incident with Chiellini was the third time that Suarez had bitten an opposing player during his professional career.
Suarez was banned for seven matches during his time at Ajax for biting PSV's Otman Bakkal in 2010, before receiving a 10-game sanction for doing the same to Branislav Ivanovic in Liverpool's Premier League match against Chelsea in April last year.
