The 22-year-old Guinea global midfielder is "understood to be keen on a move to Anfield now" rather than next summer, says the Daily Mail, even though the terms of his contract stipulate that his release clause comes into play at the end of next season.
Keita, who joined Leipzig from Red Bull Salzburg last summer, hit eight goals and seven assists last season, and Liverpool have been linked with a club-record deal to sign the Guinean global this summer.
Last week, Guinean journalist Amadou Makadji claimed Klopp had spoken to Keita about a possible move to Merseyside, before the Reds approached Leipzig on Sunday evening.
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Leipzig are set for their Champions League debut next season having finished behind Bayern Munich but ahead of Borussia Dortmund in the Bundesliga. Losing Keita would be a step back.
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Klopp believes Keita's box-to-box style will add dynamism in midfield to help unlock teams who play defensively against the Reds.
There's no denying Keita has the quality to thrive in most settings, but moving to a new country and a new league would represent a culture shock for any player.
"Even if people didn't know too much about him, they (must have) watched a few videos and they are impressive".
Goal understands Liverpool will need to smash the Bundesliga transfer record (€75m/£66m) in an attempt to prise Keita away from RB Leipzig, but their public rhetoric has remained unyielding. Convincing the German side to cash in will prove to be a massive challenge, though.