The Uruguay international was linked with a move away from the Anfield club throughout the close-season, with Premier League rivals Arsenal having two bids rejected in the transfer window.
But Liverpool fended off the interest and the 26-year-old made his first appearance of the season in Wednesday's 1-0 League Cup defeat to Manchester United, after sitting out the remainder of a 10-game suspension for biting Chelsea's Branislav Ivanovic in April.
And Ayre insisted the club never had any intention of selling their star player, and is hoping to begin talks on a deal at the end of the campaign, when Suarez will have two years remaining on his contract.
Speaking to Sky Sports News, he said: "We genuinely always wanted Luis to be here this season.
"We wanted him to continue to be here, continue to contribute because he's such a brilliant player.
"We're not a selling club. We're here to win and move forward and we see Luis as an important part of that.
"At the appropriate time we will have talks with Luis and his advisers. I'm absolutely certain in the summer or approaching the next summer, we'll have a conversation with Luis."
Ayre admitted that Suarez's incident with Ivanovic had damaged the club, but he described the Uruguayan as a "street fighter" who is an important part of Liverpool's squad.
He added: "As a footballer, he's a street fighter, he's a larger-than-life character.
"Nobody is condoning any bad behaviour but it's just something you have to deal with. He is what he is, he is the character he is, and we have to try and harness that.
"He is a great team member. The players are supportive of him and he is supportive of them. The most important thing is that he is on the football pitch, and most importantly committed to playing for Liverpool."
