Suarez was found guilty of racially abusing Evra during the corresponding fixture at Anfield last October and the row was further stoked by his failure to shake Evra's hand before the return match in February. Saturday's game at Anfield is already certain to be a highly emotional affair in the wake of last week's damning revelations concerning the Hillsborough disaster.
Ayre told talkSPORT: "We will be doing a handshake as there always is and, as regards to the Luis Suarez-Patrice Evra thing, I think it's just a non-story for this day. Despite what the media says, nobody has gone and told Luis Suarez what to do."
He added: "Luis is a grown-up, as I'm sure is Patrice Evra, and I think those players, like every other player on the pitch that day and everybody in the stadium, will recognise this isn't a day to talk about who is going to shake whose hand.
"This is about everybody being together for a much more important cause."
The club understandably want the focus to be elsewhere on what will be a poignant afternoon, the first game at Anfield since the publication of the Hillsborough Independent Panel's report into the 1989 tragedy in Sheffield.
There have been repeated calls from Liverpool and United, with the latter's manager Sir Alex Ferguson particularly vocal, for fans on both sides to show respect, especially with regard to past tragedies.
Tasteless chants about Hillsborough and the 1958 Munich air disaster have been heard in the past and the clubs want it stopped.
Ayre said: "It will be an emotional and big game for everybody. Obviously this is our first home game since the announcement and findings and it's a great opportunity for our fans to pay their respects and in many ways for everybody to pay respect to our fans.
"As much as the families and everyone else fought for this, it goes without saying that our fans have stuck with them and helped with that support throughout, so it's an important day for everyone."