Former Chelsea boss Andre-Villas Boas, Borussia Dortmund coach Jurgen Klopp and Wigan Athletic's Roberto Martinez are among the early favourites to succeed Dalglish, who was sacked as manager on Wednesday.
"What we want is the best, what we don't want is to choose quickly or choose because there's a time pressure," Ayre said on the club website on Thursday.
"It'll be about finding the right person who can do the best job for Liverpool Football Club."
Angry fans bombarded social media sites and phone-ins when it was announced that Dalglish's tenure at the club had ended only 18 months into his second spell as manager, echoing fan reaction when popular boss Rafa Benitez left the club in 2010.
Despite an eighth-place Premier League finish and a series of disappointing forays in the transfer market, the 61-year-old Scot retained the faith of swathes of supporters who saw green shoots of recovery in the club's successful cup campaigns.
Dalglish, who won three league titles in his first spell as manager between 1985 and 1991, travelled to Boston earlier this week to meet the club's principal owner John Henry and chairman Tom Werner.
His summoning and his subsequent sacking have drawn the ire of fans, who feel he deserved at least another year to improve the club's flagging results.
"I fear Liverpool's American owners have made a mistake by getting rid of Kenny Dalglish as manager," former Liverpool defender and television pundit Mark Lawrenson said in the Liverpool Daily Post.
"Twitter almost went into meltdown when it emerged Dalglish had flown over to Boston for showdown talks with Fenway Sports Group.
"I didn't think Dalglish would have gone to Boston and be sacked. If John Henry had wanted to remove someone from his position, I'd have thought he'd fly across the pond to do it to the person's face, such as with [former Director of Football Damien] Comolli. I guess I was wrong."
Dalglish's popularity could become a millstone for any new manager, who could find himself swiftly under pressure if results begin badly.
