Sources claimed on Wednesday that the Merseysiders had seen a second offer of around £66m rejected by the German club, though Klopp refused to be drawn on the matter after his side's 2-0 win over Crystal Palace in Hong Kong, simply asking "are we in Vegas?"
Nevertheless, with Keita believed to be Liverpool's top target this summer, the club have not been put off by the lack of success so far.
'Sportsmail' reports that the Anfield-based side are already putting the finishing touches on a fresh bid for the Guinean international, who is believed to be keen on the move.
The 22-year-old has established himself as one of the best midfielders in the Bundesliga after joining Leipzig from sister club Red Bull Salzburg last summer.
His all-action style is considered a perfect fit for Jurgen Klopp's famous 'gegenpressing'. As a box-to-box midfielder, critics believe that he is capable of playing in any of the three midfield positions in the German's favoured 4-3-3 system.
Keita scored 8 goals and provided a further 8 assists in 31 Bundesliga games last season as Leipzig defied the odds to qualify for the Champions League.