"We will be reminding him of the standards we expect from our players."
The head of English Professional Footballers' Association said that Suarez needs help controlling his temper.
"We have to work hard on anger management now. We have trained counselors in this field and we will be offering their services to Liverpool and the player to try to improve matters," PFA chief executive Gordon Taylor said.
Suarez is one of six nominees for the PFA player of the year award -- voted for by his peers -- which will be announced on Sunday.
"His reputation for controversy continues to increase when he can be such a good player, one of the world's best, so it is a real dilemma now," Taylor told Sky Sports.
Liverpool's American owners were widely criticized for not taking stronger action after the Suarez-Evra incident, and for allowing former team manager Kenny Dalglish to publicly defend his player while the FA was still investigating and after the verdict was announced.
Ayre said he had been in contact with the Fenway Sports Group, which also owns baseball's Boston Red Sox, about the best way to handle this situation.
"With any incident like this, any major incident at the club, we're in direct dialogue with the owners, always. I spoke to them last night and they were happy with the way we were handling the matter," he said Monday.