The 59-year-old Klopp - who was a popular coach at Liverpool for nine years until 2024 - is the favourite to take over if Nagelsmann is fired.
Bild reports on Thursday that Julian Nagelsmann has been offered seven million euros, around 77 million kronor, by the German federation in compensation for stepping down.
Nagelsmann has a contract until 2028 - but said after the loss to Paraguay in the round of 16 that he has no plans to resign.
Jürgen Klopp is global head of football at the Red Bull Group and, according to Kicker, has no exit clause in his contract. However, there is a tacit agreement that he can leave if he is offered the job of national coach.
