Former Borussia Dortmund boss Jurgen Klopp is Liverpool's first choice to replace Brendan Rodgers, who was sacked after Sunday's 1-1 draw at Everton.
The Northern Irishman was dismissed following almost three and a half years in charge, with the Reds in 10th position in the Premier League.
No deal has yet been done with Klopp, 48, who won two German Bundesliga titles with Dortmund.
Ex-Real Madrid boss Carlo Ancelotti, 56, is also being considered.
"The search for a new manager is under way and we hope to make an appointment in a decisive and timely manner," said a Liverpool statement announcing the departure of 42-year-old Rodgers.
The replacement, when he is appointed, will work within the existing structure and what has become known as Anfield's "transfer committee".
It is the group that plots and carries out transfer strategy and up until Sunday night consisted of Rodgers, scouts Dave Fallows and Barry Hunter, the man in charge of analysis Michael Edwards, FSG's Anfield representative Mike Gordon and chief executive Ian Ayre.
One of the big reasons for making the change was that owners FSG firmly believe someone else can get better results with the existing players and within that structure. In other words, they believe it is a squad that can and will perform better under someone other than Rodgers.
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Liverpool have won only four of their 11 games in all competitions this season, including victory on penalties against League Two Carlisle at home in the Capital One Cup.
Rodgers, who took over in June 2012, led the Reds to second place in the Premier League in 2013-14. They were top of the table with three games remaining, before losing out to Manchester City.
|Former Liverpool defender Mark Lawrenson|
|Liverpool need a manager who is big enough to handle the pressure, without thinking he is bigger than the club.|
|Whoever it is will have to buy into the mindset of the American owners - they want success, but their model is to try to get that by signing younger players they can train up and, sometimes, sell on.|
|Jurgen Klopp is the favourite to get the job and he would fit the bill.|
|At Borussia Dortmund he built a team that were in your face, with good players that he was making better but he also knew he would have to sell one or even two every year.|
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Liverpool are owned by Fenway Sports Group and a joint statement from principal owner John W Henry, chairman Tom Werner and president Gordon said: "We would like to place on record our sincere thanks to Brendan Rodgers for the significant contribution he has made to the club and express our gratitude for his hard work and commitment.
"All of us have experienced some wonderful moments with Brendan as manager and we are confident he will enjoy a long career in the game.
"Ambition and winning are at the heart of what we want to bring to Liverpool and we believe this change gives us the best opportunity to deliver it."
Rodgers said this week that he was "not worried" about the possibility of the sack, despite Liverpool's poor start to the season.
After the draw with Everton, he spoke of a "constant rebuilding" job and highlighted the impact of losing "four real catalysts" of his side - Jamie Carragher, Luis Suarez, Raheem Sterling and Steven Gerrard.
'This is what Klopp likes, what he needs'
Former Germany midfielder Stefan Effenberg believes Klopp will be back coaching "very, very soon" and thinks he is suited to Anfield.
Effenberg, who played 35 times for Germany, is a good friend of the former Borussia Dortmund coach and told BBC World Service Sport: "He gave me this answer a couple of weeks ago. He said, 'I'm ready for a team that's not on the highest level, to create something, to build something up'. This could be Liverpool, right?"
Speaking at the Aspire For Sport conference in Berlin he added: "Liverpool is one of the greatest atmospheres. It's pretty much the same as Dortmund. The fans stay with everything behind the club and this is what Jurgen Klopp likes and what he needs.
"So maybe, very soon, we'll hear something from Jurgen Klopp, to make a decision, maybe, for Liverpool."
|Analysis by chief football writer Phil McNulty|
|At the heart of Rodgers's demise at Liverpool was a failed, flawed recruitment policy that saw nearly £292m spent since his arrival in the summer of 2012 - but most of the world-class talent he possessed walk out of the door.|
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