Wigan manager Roberto Martinez was given permission by his club on Thursday to speak to Liverpool about filling the job vacated by Kenny Dalglish's dismissal.
The approach came while Wigan chairman Dave Whelan was with Martinez, a day after Dalglish was fired following Liverpool's disappointing season in which it finished eighth in the Premier League.
But Martinez is not Liverpool's only managerial option, with the club making inquiries about at least four managers based in England and overseas. Andre Villas-Boas, who was fired by Chelsea in March, has also been widely linked with the Anfield job.
"What we don't want is to choose quickly or choose because there's a time pressure. It'll be about finding the right person who can do the best job," Liverpool managing director Ian Ayre said.
"We have to go through a process where we assess them and understand more than what they have achieved on the pitch," Ayre added.
"What are their characteristics? How are they with players and all the different elements of a football club? It is not as simple as looking at an individual who has achieved something and say, 'He is the right guy.'"
The 38-year-old Martinez has impressed during his three years at Wigan, presiding over a series of late victories this season - including over Manchester United - to keep the northwest team in the Premier League.
"I did promise Roberto that when a big club comes he will have permission to talk to them," Whelan told British broadcaster Sky Sports.
"They don't come any bigger than Liverpool, so I gave him permission and he will be talking to them soon. I don't know when.
"The decision would be entirely Roberto's. We did speak about Liverpool. Liverpool has undergone some changes in the last two years, some massive changes, and I still think they need to settle down as a football club . Liverpool is functioning without a heart. I mentioned that to Roberto and I think it's true."
Villas-Boas is even younger than Martinez, at 34, but proved himself at Porto by winning four trophies in his only season there before being lured a year ago to Chelsea, who dumped him in March.
"His footballing philosophy and his style of play fit. He proved he could win things at Porto," former Liverpool winger John Barnes said. "He was at the right club at the wrong time with Chelsea. There were a lot of problems for him there because of the older players. He is a proven manager and he would be coming in to a humbler squad than the one at Chelsea if he came to Liverpool.
"He will get backing from the players. They will be eager to learn and listen to his plans. I think if it is going to be him he could do a good job at Liverpool."
Liverpool's priority is re-establishing itself as a Premier League force by sealing a return to the top four and Champions League qualification.
The American owners decided that Dalglish wasn't the right manager to take the club forward after an end-of-season review this week, with the 61-year-old Scot fired after 16 months in charge.
Dalglish, who won eight league titles as a player and coach before leaving in 1991, was brought back to Liverpool in January 2011 shortly after the Fenway Sports Group takeover. Following a 20-year absence, Dalglish replaced Roy Hodgson with the club hovering above the relegation zone.
After lifting Liverpool from its perilous position to sixth place, Dalglish was given a three-year contract at the end of the 2010-11 season.
But winning the League Cup to end Liverpool's six-year trophy drought in February and reaching the FA Cup final wasn't enough to convince the American owners that the club was on the right path under Dalglish.
Dalglish is the latest senior figure to leave Anfield in recent weeks, following the exit of the director of football, head of sports medicine, and communications chief.
"They were decisions made because of this plan to go forward, because there was a belief those individuals weren't good enough to take it forward at that time," Ayre said.
"The idea that all these people are leaving and there is a crisis is nonsense."