The Uruguay striker appeared to sink his teeth into the arm of Ivanovic during Sunday's 2-2 draw, and he has since apologized and been fined by the club.
Many pundits have called for the Reds to sell their star striker, while manager Brendan Rodgers admitted that "anybody can be replaced" in his postmatch observations.
But when questioned on Liverpool's official website, Ayre, who also believes the club have dealt with the issue sufficiently, refuted suggestions that Suarez could be shown the door.
"Not at all," he responded. "It affects his future in the sense that we have to work with him on his discipline - but Luis is a very important player to the club. He's a very popular player with his teammates. As we keep saying, he signed a new four-year contract last summer and we'd all love to see him here throughout that contract."
He added, "He's a fantastic player, top scorer and everything we'd want in a striker, so there's no change there. This is more about getting him back on the right track and it's largely down to Brendan now to work with him on that side of his character."
Liverpool received criticism from all quarters over their handling of last season's race row involving Suarez and Manchester United left back Patrice Evra, but Ayre believes the club have done as much as they can this time around.
"I think the most important thing is that we acted swiftly [Sunday]," he added. "Luis issued his apology and then we spoke with him last night and then again this morning. We've taken action to fine Luis for his actions. Brendan has spoken to him and I've spoken to him, and Brendan will be working with him further on his discipline.
He added, "You can see when you speak to him how sorry he is about it and he's certainly shown quite a lot of contrition to us - and as part of that, he's also asked we donate the fine to the Hillsborough Family Support Group.
"I think he felt like he let a lot of people down yesterday. We'll work with Luis - Brendan particularly - on this side of his character in his game. Hopefully that puts the matter to rest from our point of view and we'll wait and see if there's any further action from the football authorities."