Former Chelsea boss Andre-Villas Boas, Borussia Dortmund coach Juergen 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."
We study the possible replacements for King Kenny at Anfield (Note - odds correct at 8am Thursday morning):
Roberto Martinez (4/1): With Wigan finishing the season in some style by assembling seven wins in their final nine outings - including successes against Manchester United, Newcastle United and Arsenal and Liverpool - perhaps it is no surprise Martinez is the early favourite for the post. Likes to see his team get the ball on the deck and play. His style of football would appeal to the Liverpool fans, but does he have enough of a pedigree to tempt Henry?
WIgan chairman Dave Whelan seems to appear every summer extolling the benefits of Martinez. He backs him for a top job in the future. Martinez may well feel his time has come now after rejecting Aston Villa a year ago. Incidentally, Martinez is also touted again with a move to Villa. He is also Spanish, a country that is the flavour of the month at the moment in world football.
Rafael Benitez (8/1): Liverpool went back in time and got stung by Dalglish's appointment. Would they produce a similar stunt by bringing Benitez out of his self-imposed absence from the coaching game? The former Valencia manager certainly knows how to get the job done in Europe. He led Liverpool to the Champions League in 2005 with that fabled success against AC Milan on penalties after a 3-3 draw. They lost to 2-1 to Milan in the final rematch two years later.
Also picked up the FA Cup, FA Community Shield and European Super Cup, but never quite got a handle on what was needed in the Premier League with second spot in 2009 - four points behind Manchester United - being the highlight of his six-year tenure. Left the club a year after signing a five-year contract with a £6 million settlement after limping home in seventh place. Replaced Jose Mourinho at Internazionale in June 2010, but was sacked in December of the same year only days after winning the Club World Cup with Inter sixth in Serie A, 13 points off the summit of the table. Would he find Henry better to work for than Massimo Moratti at Inter?
Andre Villas-Boas (9/4): If managers had a transfer fee, young Andre would top the list. The Portuguese coach followed in the footsteps of Jose Mourinho by leading Porto to European success in 2011 before Chelsea owner Roman Abramovich paid £13.3 million to entice him to take on the Chelsea post at the ripe old age of 33. Won the Portuguese Supercup, the Primeira Liga, the Portuguese Cup and the Europa League, but found life decidedly tricky having worked under Mourinho during his previous spell with Chelsea.
With rumours of a rift between Villas-Boas and the senior players in the squad, Villas Boas was dismissed in March after a 3-1 loss to Napoli in the first leg of a Champions League last-16 tie. Chelsea were fifth in the table when Villas-Boas was dismissed. They finished sixth behind Newcastle United. Has been on holiday since then, but the Juan Mata doppelganger may feel the time is right to return if Liverpool come calling.
Brendan Rodgers (7/1): Like Villas-Boas, the Northern Irishman has been a career coach having retired as a player due to injury at the age of 20. Like Villas-Boas, has also enjoyed the ups and downs of life in management in his thirties. Left Watford after they finished 13th in the Championship before departing Reading after a poor sequence of results in December 2009, but has found the right chemistry at Swansea. Led them to the Championship play-off final where they somewhat ironically defeated Reading at Wembley. Led Swansea to 11th place in the table.
They beat Liverpool 1-0 on the final day. Rodgers is a self-confessed Celtic fan who has signed a three-and-a-half-year contract to continue managing the Welsh club, but that will mean little if Liverpool want to speak to him. His side play a lovely, eye-catching style of football. Like Martinez, Rodgers, 39, is a personable, modern coach who is also adept at handling public relations.
Martin O'Neill (20/1): If ever there was a figure with a CV built for a blue-chip job, it is surely the Sunderland manager. Led Leicester to the League Cup in 1997 and 2000 before winning three Scottish Premier Leagues, three Scottish Cups and a Scottish League Cup with Celtic. Led Celtic to a win over Liverpool in the last eight of the 2003 Uefa Cup before losing to 3-2 to Jose Mourinho's terrific Porto team in a memorable final.
Spent four years at Aston Villa, but failed to break through the ceiling of the top four before departing two years ago apparently over his (lack of) budget for new players. Was linked with the England post and Liverpool job before Roy Hodgson left Fulham for Anfield. Arrived at Sunderland in December 2011 and lifted them out relegation trouble. They finished the season in 13th place. Having just turned 60, would be difficult for O'Neill to say no if Liverpool are prepared to offer him decent money to spend.
David Moyes (50/1): Okay, okay...we know. It is highly unlikely Moyes would betray the blue half of Merseyside by jumping ship for Liverpool. Moyes would probably laugh off the notion, but why should he not be linked with such a post? The Scotsman has worked a minor miracle at Everton on a low budget since arriving in 2002. His latest body of work saw Everton finish above Liverpool in the Premier League standings in seventh position.
Perhaps he is waiting for the Manchester United job to become available. Or perhaps he will continue to impress at Everton. There is little doubt that he deserves some money to spend. Imagine where Everton could go if Moyes had the £110 million Dalglish spent on players last year?
Other selected odds: Fabio Capello (12/1), Frank Rijkaard (12/1), Harry Redknapp (20/1), Jurgen Klopp (20/1), Didier Deschamps (25/1), Bob Bradley (33/1)
Who do you think should be the next Liverpool boss? Is there another candidate you believe they should consider? Let us know your thoughts below.