Klopp is said to be preparing a move for his former playmaker. The Germany international has been in impressive form so far this term, as Dortmund have started the Bundesliga campaign well.
Brendan Rogers was recently sacked as Liverpool manager and Germany coach Jurgen Klopp was recently appointed as his replacement.
Klopp has signed a three-year deal worth £15 million which works out at £5 million per-season but he could earn up to £7 million with bonuses included.
Dortmund rejected an approach from United just before the end of the summer window for Gündoğan according to the German newspaper Express.
Klopp won two Bundesliga titles and finished the 2012-13 Champions League as runners-up during his time in charge of Dortmund.