Since arriving at Anfield in 2015, Klopp has guided Liverpool to 155 victories from 256 games in charge, and the 52-year-old is committed to the Champions League holders until 2024.
However, according to The Mirror, Klopp has privately admitted that Liverpool is likely to be his last managerial role at club-level, and he is said to have taken an interest in the national team role.
The report also claims that the former Borussia Dortmund manager has no intention of leaving Anfield before 2022, when Low's deal with the national side is set to expire.
Liverpool's owners Fenway Sports Group are said to be keen on creating their own Sir Alex Ferguson-like era with Klopp at the helm, but they may face an uphill battle to secure the long-term future of the German.
Low has taken charge of 181 games for Germany since 2006, winning the 2014 World Cup but failing to advance past the group stage four years later.