German newspaper Bild has reported that the 47-year-old has asked Dortmund chief executive Hans-Joachim Watzke to release him early from his contract.
Klopp, regarded as one of the most talented coaches in European football, has been in charge of Dortmund since 2007, winning two Bundesliga titles and guiding the club to the Champions League final in 2013.
This season has been more of a struggle, though, with Dortmund fighting at the bottom end of the league for much of the campaign. They currently sit 10th.
The club will hold a press conference at 1230 (BST) on Wednesday.
According to Bild, Klopp will take some time out before returning to management.
He has long been linked with managing in the Premier League.