The exchange came during MagentaTV's coverage of the World Cup's opening match on Thursday, where Klopp and Thomas Muller are working as television pundits.
Discussing the team Germany should select against Curacao, Klopp made a comment that referenced Nagelsmann's position.
"Fortunately Julian Nagelsmann picks the team," Klopp said, before adding a pointed "for now."
Muller, who played under Nagelsmann at Bayern Munich, responded immediately on air.
"Kloppo. It's June. You're already in September," Muller said.
The exchange amused both pundits, with the implication being that Nagelsmann's position could be under threat by September despite his contract running until after Euro 2028.
Klopp is widely regarded as a leading candidate to take over as Germany head coach should Nagelsmann leave the role after the World Cup.
Speaking at a press conference in Houston ahead of Germany's opening match against Curacao, Nagelsmann was first asked about the comments by a German reporter.
"Right, next question," Nagelsmann said, declining to engage further.
An English-language tabloid journalist returned to the subject, prompting a fuller response from the head coach.
"Are you from the same publication?" Nagelsmann joked, before switching to English to address the question directly. "I think it's not the right way for me to talk about this topic," he said.
"I think it's important that everyone assesses the situation for themselves. I have my opinion and my point of view, but I will not tell you."
Nagelsmann, 38, did offer praise for one of the other television pundits working at the tournament.
Lothar Matthaus, Germany's record international, had backed Nagelsmann's decision to deploy Jamal Musiala as a central midfielder rather than further forward.
Klopp and Muller had separately suggested injured Bayern forward Deniz Undav could have played in a more advanced role, a suggestion Klopp later softened.
Germany's World Cup campaign began against Curacao in Houston on Sunday evening, before fixtures against Ivory Coast in Toronto and Ecuador in New Jersey complete their group schedule.
For Nagelsmann, the on-air sparring from two of Germany's most prominent football figures is unlikely to be the last word on his future, regardless of how he chooses to address it in public.