Rodgers had refused a request to meet with Liverpool following Kenny Dalglish's sacking earlier this month, but the Reds made a renewed effort to speak to the former Reading and Watford chief after holding discussions with Wigan's Roberto Martinez last week.
The 39-year-old Northern Irishman is believed to have spoken to Liverpool officials on Wednesday and the BBC reports the Merseyside giants are ready to pay between £4 million ($6.2mn) and £5 million ($7.7mn) in compensation to Swansea to secure his services.
Reports that Rodgers has agreed a three-year contract with Liverpool are described as "categorically untrue" by the Anfield club, but it seems he is now well on course to land one of the biggest jobs in English football.
Rodgers is regarded as one of the Premier League's brightest young managers after guiding unheralded Swansea to an impressive 11th place finish on a relatively meagre budget just one season after securing promotion from the Championship.
When Liverpool first approached Rodgers, it was claimed he was unhappy that the club's American owners wanted to conduct interviews with a host of potential candidates as it would leave him having to return to Swansea with his position undermined if he failed to get the job.
Martinez had been the early favourite after meeting with Liverpool's principal owner John Henry in Miami last week, but the Spaniard was said to be concerned about the Reds' desire to hire a director of football, with former Ajax boss Louis van Gaal reportedly in the running for that role.
Wigan chairman Dave Whelan claimed Martinez had been offered the job, but Liverpool never confirmed that and now appear to have turned their sights on Rodgers.
If Rodgers does get the job, it will complete a remarkable rise following his sacking as Reading boss just three years ago.
Since then Rodgers, who regards Jose Mourinho as his mentor after working as the Portuguese coach's reserve-team boss at Chelsea, has completely rehabilitated his reputation.
Swansea's eye-catching style of play earned a standing ovation from Liverpool fans after a 0-0 draw at Anfield in November and Rodgers also masterminded a 1-0 win over the Reds on the final day of the season, as well as beating Manchester City and Arsenal at the Liberty Stadium.
He would take over a Liverpool squad at a low ebb after finishing eighth in the Premier League, 17 points away from the Champions League spots that Henry had targeted at the start of the season.
The Reds did win the League Cup under Dalglish, but their season ended with a lacklustre 2-1 defeat in the FA Cup final against Chelsea and the new boss will be charged with reinvigorating team sorely in need of an infusion of creativity and pace.
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