The report states that the Red’s boss has held informal talks with the 25-year-old midfielder.
The Germany international worked alongside Klopp at Dortmund for several years before Klopp left at the end of last season.
Gundogan admitted earlier this week week that he was considering his future.
“There is still no decision, and of course, everything has to be considered very carefully, because your career is limited in time. But to be honest, everything is still open,” Gundogan was quoted as saying by Goal.
Gundogan has been linked with a move to Anfield since the appointment of Jurgen Klopp as manager in place of Brendan Rodgers. Dortmund rejected an approach from Liverpool’s rivals Manchester United earlier on in the season according to the German newspaper Express.
The midfield playmaker is tied into a contract at Dortmund until 2017 after signing a contract extension at the start of the season.