The two clubs go head-to-head in the FA Cup this weekend, with the winner booking a place in the fifth round.
But this will be the first time that the two clubs play each other since Luis Suarez was banned for eight matches for racially abusing Patrice Evra back in October, so tensions will be high.
Ayre told BBC Radio Merseyside: "We mustn't detract from fans chanting and singing at each other but it needs to be a rivalry thing rather than a hatred thing."
Sir Alex Ferguson has written to travelling away fans to behave at Anfield and Ayre has appealed to both sets of fans to understand the nature of the game, but to not let things turn ugly.
He said: "It is the FA Cup, a great competition in English football and one we have a great history in and we want that to be what everyone focuses on.
"We have massive rivalry with Manchester United but those who are respectful also have respect for them because they are a great footballing side and team and, like us, they have been very successful.
"It is about a great spirit and a match between two great teams and it is certainly not hatred - there is no place for hatred in football.
"The banter on the terraces is great, both managers feel that as do the players. Far from trying to neutralise it we want our fans to be as vocal as they normally are against Manchester United.
"We definitely don't want to detract from that but both clubs have histories which have some things in people don't want to hear being sung about and so we want the fans to get behind the team, have a bit of fun and enjoy the game."