Gündoğan has started the season well in the Bundesliga under Thomas Tuchel.
La Liga champions Barcelona are also understood to be interested in securing his services of the Germany international.
The former Manchester United target has helped Dortmund to second in the table after eight games of the new season.
Klopp has been appointed as the Liverpool manager on a three-year contract in place of Brendan Rodgers.
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.