Goal exclusively revealed on July 10 that the Gunners were preparing to make a £35 million bid for the 26-year-old after contacting the Anfield club regarding his availability.
Arsene Wenger recently refused to answer questions on Arsenal's interest in Suarez but Ayre has now revealed that Liverpool have rejected a bid from the Gunners, with the striker yet to confirm his desire to leave.
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"I've heard him say it would great to have an opportunity to play for Arsenal and likewise Real Madrid. But he certainly hasn't called me up and said he wants to leave.
"We get offers for lots of our players at different times. Unless we get to a situation where that player absolutely doesn't want to be here and we don’t want him to be here then that's a different kettle of fish.
"Right now we'd love to see Luis put on a Liverpool shirt for this season and beyond and we hope that once he gets back things will settle down."
Ayre also insisted that Liverpool are determined to keep hold of the former Ajax man, who scored 30 goals last season, and will be able to meet his ambitions of playing Champions League football in the future.
"This is an ambitious young player, he’s talked in the media about wanting to play in the Champions League and all these things," Ayre added. "It’s our job to convince Luis that this is the right place to achieve those things."
"I don’t think anyone will be against him at this point. He only needs to do what he did last season and everyone will feel he’s in the right place and he should carry on getting the support that he deserves and gets from Liverpool.
"We’d love to see Luis put on a Liverpool shirt for this season and beyond and we hope that once he gets back things will settle down."