The Reds agreed last August a deal in excess of £50m that would see Keita spend another season in the Bundesliga before moving to Anfield in the summer.
However, Jurgen Klopp has been exploring the possibility of moving forward the Guinea international midfielder's arrival to this transfer window following the £142m departure of Philippe Coutinho to Barcelona.
Leipzig have been resistant to that suggestion given they are challenging for Champions League qualification and are preparing for the Europa League knockout stages.
But according to German news outlet Bild, the Bundesliga side are now softening their stance.