Liverpool had explored the possibility of getting him sooner but following Coutinho's departure, Klopp is now eager to draft him in ahead of the second half of the season.
The Reds have already agreed a deal in excess of £50million to sign the Guinea midfielder on July 1, with the deal set to see the Reds pay a transfer premium in addition to his release clause, dependent on how Leipzig fare this season.
Keita has established himself as a top midfielder as part of an RB Leipzig side that has soared to relevance following their promotion to the Bundesliga in 2016.
RB could be playing "transfer" poker with Liverpool and trying to get as much out of the early move as possible.
The report states that the Bundesliga outfit would consider allowing the Guinea worldwide to leave early for between £13m and £17.5m.
The German club is not short of funds owing to energy drinks company Red Bull's backing, but are said to want to raise funds to invest to keep with Uefa's Financial Fair Play regulations.
However, those beliefs were initially quashed by Leipzig CEO Oliver Mintzlaff who said: "Neither Naby nor his agent have made the request to leave the club in the winter".
Liverpool have money to spend after they sold Philippe Coutinho to Barcelona during the weekend for a fee believed to be in the region of £145 million.
The two clubs agreed a €55m deal for the Guinea global to move to Anfield after the World Cup finals in Russian Federation this summer.
Liverpool FC are now preparing for their crunch Premier League clash against Manchester City at The Etihad on Sunday afternoon.