Klopp is understood to be open to the idea of a move, the reports said, adding that talks between Liverpool and his representatives were progressing well.
The 48-year-old has been out of work since leaving Dortmund in the summer.
Klopp won two German Bundesliga titles at Dortmund and led the club to the finals of the Champions League in 2013, but lost the German Cup final to Wolfsburg in his final game in charge at the end of the 2014-15 season during which his side struggled in the domestic league, eventually finishing seventh.
Klopp is understood to have spoken to former Liverpool and Germany midfielder Dietmar Hamann about the club, the city and the fans, the BBC reported.
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Former Germany midfielder Stefan Effenberg believes Klopp will be back coaching "very, very soon" and thinks he is suited to Anfield.
"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?," Effenberg, who has played 35 times for Germany, told the BBC.
"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," the BBC quoted Effenberg as saying at the ASPIRE4SPORT conference in Berlin.
"So maybe, very soon, we'll hear something from Jurgen Klopp, to make a decision, maybe, for Liverpool."
In a statement issued by the League Managers' Association on his behalf, Rodgers, who was sacked after Sunday's 1-1 Merseyside derby draw at Everton, said he was 'incredibly disappointed' his 3-1/2 year tenure at Anfield was over.
Negotiations will now take place over his severance package, which could reportedly cost Liverpool in excess of seven million pounds.
(Reporting by Simon Jennings in Bengaluru; editing by Sudipto Ganguly)
