Since arriving at Anfield in January 2011, the controversial Uruguayan has been punished for racially abusing Manchester United defender Patrice Evra, as well as receiving a 10-game ban for biting the arm of Chelsea's Branislav Ivanovic.
The 26-year-old made his return to action from suspension by starting last night's 1-0 defeat to Manchester United in the third round of the Capital One Cup.
According to The AP, Ayre described Suarez as a "street fighter" during a talk at the Sport Industry breakfast in London, and has insisted that the forward has "responded well" following the ban.
However, Ayre has criticised Suarez's previous conduct by saying: "Any types of incident of that nature are damaging to the brand."
Suarez publicly expressed a desire to leave Anfield during the summer, but the Merseyside outfit blocked a move away from the club.